qu.1.topic=VoR:Ver about x=a@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=VoR x,x^2,x=$aLeft@
qu.1.1.comment=<p><img width="308" height="260" src="__BASE_URI__pictures/xsquared2.png" alt="" /></p>
<p>This is a shell.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math></p>
<p>thickness <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-5,-1);
$b=-$a;
$pa=plotmaple("plot([x^2,x],x=$a..2,y=-1..2,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.1.1.uid=40d2acff-c156-4b66-90c8-5963303bf821@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolutions;
  Sub-Topic=Setup;
@
qu.1.1.question=<p>$pa</p>
<p>&nbsp;The integral using vertical rectangles which finds the volume obtained when the region bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover></mrow><mrow><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&InvisibleTimes;</mo><mo    mathvariant='normal'             >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.1.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.1.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mo    mathvariant='normal'             >&sdot;</mo></mrow><mfenced><mrow><mrow><mi    mathvariant='italic' >x</mi></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></math>@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mi    mathvariant='normal' >$a</mi><mn    mathvariant='normal' >1</mn></munderover></mrow><mfenced><mrow><mrow><mi    mathvariant='italic' >x</mi></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=VoR x,x^2,x=$aRight@
qu.1.2.comment=<p><img width="286" height="260" src="__BASE_URI__pictures/xsquared.png" alt="" /></p>
<p>This is a shell</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup></mrow></mstyle></math></p>
<p>thickness<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(1,5);
$pa=plotmaple("plot([x^2,x],x=-1..$a,y=-1..2,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.1.2.uid=bdb43f56-f1b4-4ffc-826f-75b800b8fa14@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolution;
  Sub-Topic=Setup;
@
qu.1.2.question=<p>$pa</p>
<p>&nbsp;The integral using vertical rectangles which finds the volume obtained when the region bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.1.2.answer=1@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover></mrow><mrow><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&InvisibleTimes;</mo><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.1.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo></mrow><mfenced><mrow><mrow><mi    mathvariant='italic' >x</mi></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></math>@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$a</mi></munderover></mrow><mfenced><mrow><mrow><mi    mathvariant='italic' >x</mi></mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >xdx</mi></mrow></math>@
qu.1.2.fixed=@

qu.2.topic=VoR:Ver about y=a@

qu.2.1.mode=Multiple Choice@
qu.2.1.name=VoR x,x^2,y=$aAbove@
qu.2.1.comment=<p><img width="308" height="260" src="__BASE_URI__pictures/xsquared3.png" alt="" /></p>
<p>This is a difference of discs.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ly</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(2,5);
$pa=plotmaple("plot([x^2,x],x=-1..2,y=-1..$a,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.2.1.uid=7ea7e66f-979d-43bb-998b-0ce09c160eb6@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolution;
  Sub-Topic=Setup;
@
qu.2.1.question=<p>$pa</p>
<p>The integral using vertical rectangles which finds the volume obtained when the region</p>
<p>bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>is rotated about the line&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.2.1.answer=1@
qu.2.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mrow><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.2.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><msup ><mrow><mi    mathvariant='italic' >x</mi></mrow><mn    mathvariant='normal' >4</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$a</mi></munderover></mrow><mfenced><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></math>@
qu.2.1.fixed=@

qu.2.2.mode=Multiple Choice@
qu.2.2.name=VoR x,x^2,y=$aBelow@
qu.2.2.comment=<p><img width="308" height="260" src="__BASE_URI__pictures/xsquared4.png" alt="" /></p>
<p>This is a difference of discs.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Uy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ly</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>z</mi><mrow><mi>i</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-4,-1);
$b=-$a;
$pa=plotmaple("plot([x^2,x],x=-1..2,y=$a..2,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.2.2.uid=7d977a79-a56b-444f-8103-f3857291b09e@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolution;
  Sub-Topic=Setup;
@
qu.2.2.question=<p>$pa</p>
<p>The integral using vertical rectangles which finds the volume obtained when the region bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.2.2.answer=1@
qu.2.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi    mathvariant='italic' >x</mi></mrow></mrow></mrow></math>@
qu.2.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><msup ><mrow><mi    mathvariant='italic' >x</mi></mrow><mn    mathvariant='normal' >4</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mn    mathvariant='normal' >1</mn></munderover><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$b</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.2.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mrow><mn>1</mn></mrow></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mi>dx</mi></mrow></mstyle></math>@
qu.2.2.fixed=@

qu.3.topic=VoR:Hor about x=a@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=VoR y,sqrt(y),x=$aL@
qu.3.1.comment=<p><img width="329" height="260" src="__BASE_URI__pictures/rootx2.png" alt="" /></p>
<p>This is a difference of discs.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math></p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-5,-1);
$b=-$a;
$pa=plotmaple("plot([x^2,x],x=0..2,y=-1..1,view=[-$b..$b,-1..1],thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.3.1.uid=da511cf1-e16c-411c-969a-bc3f31b4975d@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolutions;
  Sub-Topic=Setup;
@
qu.3.1.question=<p>$pa</p>
<p>The integral using horizontal rectangles which finds the volume obtained when the region bound by&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.3.1.answer=1@
qu.3.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi><mrow><mi></mi></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mn>2</mn><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.1.fixed=@

qu.3.2.mode=Multiple Choice@
qu.3.2.name=VoR y,sqrt(y),x=$aR@
qu.3.2.comment=<p><img width="329" height="260" src="__BASE_URI__pictures/rootx.png" alt="" /></p>
<p>This is a difference of discs.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>R</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,5);
$pa=plotmaple("plot([x^2,x],x=-1..$a,y=-1..2,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.3.2.uid=e7bb0e63-b62d-4535-87e0-76e3e65b591f@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolution;
  Sub-Topic=Setup;
@
qu.3.2.question=<p>$pa</p>
<p>The integral using horizontal rectangles which finds the volume obtained when the region bound by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>
<p>&nbsp;</p>@
qu.3.2.answer=1@
qu.3.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>&pi;</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mrow></mstyle></math>@
qu.3.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mn>1</mn></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mstyle></math>@
qu.3.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mrow><mi></mi></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mrow></mstyle></math>@
qu.3.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi></mi></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>d</mi></mrow><mi>y</mi></mrow></mstyle></math>@
qu.3.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mi mathvariant='normal'>$a</mi></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>y</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi><mrow><mi></mi></mrow></mrow></mstyle></math>@
qu.3.2.fixed=@

qu.4.topic=VoR:Hor about y=a@

qu.4.1.mode=Multiple Choice@
qu.4.1.name=VoR y,sqrt(y),y=aAbove@
qu.4.1.comment=<p><img width="333" height="260" src="__BASE_URI__pictures/rootx3.png" alt="" /></p>
<p>This is a shell.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ly</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p>thickness = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(1,5);
$pa=plotmaple("plot([x^2,x],x=0..1,y=-1..$a+1,view=[-1..2,-1..$a],thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.4.1.uid=8947755d-8d1f-4528-b968-4b4b24568a9c@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolutions;
  Sub-Topic=Setup;
@
qu.4.1.question=<p>$pa</p>
<p>The integral using horizontal rectangles which finds the volume obtained when the region bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.4.1.answer=1@
qu.4.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mrow></mstyle></math>@
qu.4.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mrow></mstyle></math>@
qu.4.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dy</mi></mrow></mrow></mstyle></math>@
qu.4.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>&pi;</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi></mi></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>d</mi></mrow><mi>y</mi></mrow></mstyle></math>@
qu.4.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mrow><mn>1</mn></mrow></munderover></mrow><msup><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></msup><mrow></mrow><mi>dy</mi><mrow></mrow></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>@
qu.4.1.fixed=@

qu.4.2.mode=Multiple Choice@
qu.4.2.name=VoR y,sqrt(y),y=aBelow@
qu.4.2.comment=<p><img width="308" height="260" src="__BASE_URI__pictures/rootx4.png" alt="" /></p>
<p>This is a shell.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ly</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p>thickness <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&par;</mo><mi>&Delta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&par;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math></p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(-5,-1);
$b=-$a;
$pa=plotmaple("plot([x^2,x],x=0..2,y=-1..2,view=[-1..2,-$b..$b],thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=350,width=350'");@
qu.4.2.uid=10f4d23c-8b16-4536-8573-8de3f96a95e6@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volumes of revolutions;
  Sub-Topic=Setup;
@
qu.4.2.question=<p>$pa</p>
<p>The integral using horizontal rectangles which finds the volume obtained when the region bounded by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>y</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mstyle></math>is rotated about the line <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mstyle></math> is given by</p>@
qu.4.2.answer=1@
qu.4.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mi>dy</mi></mrow></mstyle></math>@
qu.4.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mi>dy</mi></mrow></mstyle></math>@
qu.4.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mi>dy</mi></mrow></mstyle></math>@
qu.4.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mfenced open='(' close=')' separators=','><mrow><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mrow><mi></mi></mrow><mi>dy</mi></mrow></mstyle></math>@
qu.4.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mrow><mi>&pi;</mi></mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mn>1</mn></munderover></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mrow><mi></mi></mrow><msup><mi>y</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><msqrt><mrow><mi>y</mi></mrow></msqrt></mrow></mrow></mfenced><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi></mi></mrow><mi>dy</mi></mrow></mstyle></math>@
qu.4.2.fixed=@

qu.5.topic=VolumeSphere@

qu.5.1.mode=Multiple Choice@
qu.5.1.name=VoRsphere@
qu.5.1.comment=<p><img width="329" height="260" src="__BASE_URI__pictures/sphere.png" alt="" /></p>
<p>This is a disc.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><msup><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfenced><mrow><mn>2</mn></mrow></msup><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$r</mi></mrow><mrow><mi mathvariant='normal'>$r</mi></mrow></munderover><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$r=rint(1,4);
$R=$r^2;
$pa=plotmaple("plot([$r*cos(t),$r*sin(t),t=0..Pi],x=-$r..$r,y=-$r..$r,thickness=2,tickmarks=[2,2
]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=489fdc44-d608-49c3-b83a-b9c9d9ce9d95@
qu.5.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>$pa</p>
<p>By rotating the semi-circle <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mstyle></math> about the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi></mrow></mstyle></math>axis, we can find the volume of a sphere of radius <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$r</mi></mrow></mstyle></math>. The integral which gives this volume is</p>@
qu.5.1.answer=1@
qu.5.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$r</mi></mrow><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mi    mathvariant='normal' >$R</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.5.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$r</mi></mrow><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='normal' >$R</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.5.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$r</mi></mrow><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mi    mathvariant='normal' >$R</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.5.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$r</mi></mrow><mi mathvariant='normal'>$r</mi></munderover></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow><mi>dx</mi></mrow></mrow></mstyle></math>@
qu.5.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&pi;</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$r</mi></mrow><mi mathvariant='normal'>$r</mi></munderover><mrow><msqrt><mrow><mi mathvariant='normal'>$R</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow><mrow><mi>dx</mi></mrow></mrow></mstyle></math>@
qu.5.1.fixed=@

qu.6.topic=VolumeCone dx@

qu.6.1.mode=Multiple Choice@
qu.6.1.name=Volume Cone dx@
qu.6.1.comment=<p><img width="254" height="227" src="__BASE_URI__pictures/cone.png" alt="" /></p>
<p>This is a shell</p>
<p>thickness <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Uy</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Ly</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$h</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$h</mi><mrow><mi mathvariant='normal'>$r</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn>0</mn></mrow></munder><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>r</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>thickness</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></munder><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$h</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$h</mi><mrow><mi mathvariant='normal'>$r</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&Delta;x</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi mathvariant='normal'>$r</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$h</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$h</mi><mrow><mi mathvariant='normal'>$r</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$r=rint(2,5);
$h=rint(1,6);
condition:eq(gcd($r,$h),1);

$pa=plotmaple("plot([$h/$r*x,$h],x=0..$r,y=0..$h,view=[-$r..$r+1,-$h..$h+1],thickness=2,tickmarks=[4,4
]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.6.1.uid=db0b450d-8ce4-4a7d-88da-c4a66d85dfe6@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volume of revolution;
  Sub-Topic=Setup;
@
qu.6.1.question=<p>$pa</p>
<p>By rotating the line&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$h</mi></mrow><mrow><mi mathvariant='normal'>$r</mi></mrow></mfrac><mi>x</mi></mrow></mstyle></math> about the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>axis from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$r</mi></mrow></mstyle></math>, we can find the volume of a cone of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$r</mi></mrow></mstyle></math> and height <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$h</mi></mrow></mstyle></math>. The integral using vertical rectangles which gives this volume is</p>@
qu.6.1.answer=1@
qu.6.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='normal' >$h</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mfrac    ><mi    mathvariant='normal' >$h</mi><mi    mathvariant='normal' >$r</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.6.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><msup ><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$h</mi><mi    mathvariant='normal' >$r</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.6.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$h</mi><mi    mathvariant='normal' >$r</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.6.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><msup ><mfenced><mrow><mi    mathvariant='normal' >$h</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mfrac    ><mi    mathvariant='normal' >$h</mi><mi    mathvariant='normal' >$r</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.6.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$r</mi></munderover></mrow><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$h</mi><mi    mathvariant='normal' >$r</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$h</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dx</mi></mrow></mrow></math>@
qu.6.1.fixed=@

qu.7.topic=VolumeCone dy@

qu.7.1.mode=Multiple Choice@
qu.7.1.name=Volume Cone dy@
qu.7.1.comment=<p><img width="254" height="227" src="__BASE_URI__pictures/cone2.png" alt="" /></p>
<p>This is a disc.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Ux</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>Lx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>h</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>V</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>h</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>&Delta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mn>0</mn></mrow></munder><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>z</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><msub><mi>&Delta;y</mi><mrow><mi>i</mi></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi mathvariant='normal'>$h</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>y</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mstyle></math></p>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$r=rint(2,6);
$h=rint(2,6);
condition:eq(gcd($r,$h),1);

$pa=plotmaple("plot([$h/$r*x,$h],x=0..$r,y=0..$h,view=[-$r..$r+1,-$h..$h+1],thickness=2,tickmarks=[4,4
]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.1.uid=6eb4ec78-5e70-4d43-864d-2b3284a0e1fe@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Volume of revolution;
  Sub-Topic=Setup;
@
qu.7.1.question=<p>$pa</p>
<p>By rotating the line&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$r</mi><mrow><mi mathvariant='normal'>$h</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>y</mi></mrow></mstyle></math> about the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi></mrow></mstyle></math>axis from <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$h</mi></mrow></mstyle></math>, we can find the volume of a cone of radius&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$r</mi></mrow></mstyle></math> and height <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$h</mi></mrow></mstyle></math>. The integral using horizontal rectangles which gives this volume is</p>@
qu.7.1.answer=1@
qu.7.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$h</mi></munderover></mrow><mrow><msup ><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$h</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >y</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dy</mi></mrow></mrow></math>@
qu.7.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$h</mi></munderover><mi    mathvariant='italic' >y</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo></mrow><mrow><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$h</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >y</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dy</mi></mrow></mrow></math>@
qu.7.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$h</mi></munderover></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='normal' >$r</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$h</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >y</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dy</mi></mrow></mrow></math>@
qu.7.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$h</mi></munderover></mrow><mrow><msup ><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$h</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >y</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$r</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dy</mi></mrow></mrow></math>@
qu.7.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mn    mathvariant='normal' >2</mn><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >&pi;</mi></mrow><mrow><munderover  accentunder='false'><mo    mathvariant='normal'  form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mn    mathvariant='normal' >0</mn><mi    mathvariant='normal' >$h</mi></munderover><mi    mathvariant='italic' >y</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo></mrow><mrow><msup ><mfenced><mrow><mfrac    ><mi    mathvariant='normal' >$r</mi><mi    mathvariant='normal' >$h</mi></mfrac><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >y</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&InvisibleTimes;</mo><mi    mathvariant='italic' >dy</mi></mrow></mrow></math>@
qu.7.1.fixed=@

