qu.1.topic=Theory@

qu.1.1.mode=Multiple Selection@
qu.1.1.name=TheDefIntegral2@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=9e6c4a8e-c425-40f7-ac2f-58c5c4cabb24@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Sub-Topic=Theory;
  Topic=Definite Integral;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
@
qu.1.1.question=<p>Click beside each true statement.</p>@
qu.1.1.answer=1, 3, 4@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>@
qu.1.1.choice.3=When evaluating a definite integral, we do not have to add <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></math>.@
qu.1.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>P</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></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>F</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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><mi>&Delta;</mi><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>F</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow></math>@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mfenced open='&par;' close='&par;' separators=','><mrow><mi>P</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&srarr;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></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>f</mi><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><mi>&Delta;</mi><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mrow></mrow></math>@
qu.1.1.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></math> always measures the area between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> and the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> axis between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>a</mi></mrow></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>b</mi></mrow></math>.@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=TheDefIntegral@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=54dd7f5b-e2d3-4419-9faa-3e599e0b1374@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Definite Integrals;
  Sub-Topic=Theory;
@
qu.1.2.question=<p>Click beside each true statement.</p>@
qu.1.2.answer=1, 2, 3, 5, 7, 8@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>a</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>0</mn></mrow></math>@
qu.1.2.choice.3=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&amp;period;</mo></mrow></math>@
qu.1.2.choice.4=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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></math> then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mfenced open='(' close=')' separators=','><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow><mrow><mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&amp;period;</mo></mrow></math>@
qu.1.2.choice.5=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is continuous, then<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></math> exists.@
qu.1.2.choice.6=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></math> exists, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>f</mi></mrow></math> is differentiable.@
qu.1.2.choice.7=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>c</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>c</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></math>@
qu.1.2.choice.8=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></munderover><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced></mrow></math>@
qu.1.2.fixed=@

qu.2.topic=DerOfIntegral@

qu.2.1.question=<p align="left">Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>.</p>@
qu.2.1.maple=evalb(simplify(($ANSWER)-($RESPONSE))=0);@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=$ANS@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=Der(Int)F'@
qu.2.1.comment=<p>Remember<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mi>d</mi><mrow><mi>dx</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>Bottom</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi>Top</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>Top</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>Top</mi><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>Bottom</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>Bottom</mi><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&prime;</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>.</p>
<p>In our case <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow></math>$F<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$step.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=range(-5,5);
$n=range(2,10);
$m=range(2,10);
condition: ne($a,0);
$f=switch(range(7),"sin(x)","cos(x)","x^($m)+($a)", "1/(x+($a))^($m)", "cos(x+($a))","(x+($a))^($m)","sin(($a)*x)");
$T=switch(range(4),($a),"x^($n)","sin(x)","cos(x)");
$B=switch(range(4),($a),"sin(x)","cos(x)","x^($n)");
condition:ne($T,$B);
$M=maple("
Tp:=diff($T,x):
Bp:=diff($B,x):
ANS:=eval($f,x=$T)*(Tp)-eval($f,x=$B)*(Bp):
convert(ANS,string),
MathML[ExportPresentation]($Int($f,x=$B..$T))
");
$ANS=switch(0,$M);
$F=switch(1,$M);
$step=mathml("$ANS");@
qu.2.1.uid=3f721aa7-03e5-492e-9e46-63248d26bba3@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Definite Integrals;
  Sub-Topic=Derivative of definite integral;
@

qu.3.topic=Area@

qu.3.1.mode=Inline@
qu.3.1.name=BasicArea(x^2+a)@
qu.3.1.comment=<p>Area = $feedback <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$fb</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$fa</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi></mrow></math></p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$f="x^2+5";
$a=range(-3,2);
$b=range(($a)+1,5);
$F=mathml("y=$f");
$i="'int($f,x=$a..$b)'";
$M=maple("
fb:=eval(x^3/3+5*x,x=$b):
fa:=eval(x^3/3+5*x,x=$a):
ans:=fb-fa:
convert(fb,string),
convert(fa,string),
convert(ans,string),
MathML[ExportPresentation]($i = x^3/3 + 5*x)");
$fb=switch(0,$M);
$fa=switch(1,$M);
$ANS=switch(2,$M);
$feedback=switch(3,$M);
$plot=plotmaple("plot([[x,$f,x=-abs($a)-1..abs($b)+1],
[$a,x,x=0..eval($f,x=$a)],[$b,x,x=0..eval($f,x=$b)]],thickness=2,
color=[black,red,red]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.1.uid=541e7539-d113-4dfa-908e-8ceed2a67882@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Definite Integrals;
  Sub-Topic=Area;
@
qu.3.1.weighting=1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.maple_answer=$i@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.libname=@
qu.3.1.part.1.mode=Maple@
qu.3.1.part.1.allow2d=2@
qu.3.1.part.1.plot=@
qu.3.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.1.type=maple@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.maple_answer=int($f,x=$a..$b)@
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.libname=@
qu.3.1.part.2.mode=Maple@
qu.3.1.part.2.allow2d=1@
qu.3.1.part.2.plot=@
qu.3.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.2.type=formula@
qu.3.1.question=<p>Below is the graph of $F.</p><p>$plot</p><p>(a) The integral which gives the area bounded by $F and the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>axis from <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math> is</p><p>&nbsp;</p><p><1><span> </span></p><p>(<strong>In the equation editor, select <img width="54" height="41" alt="" src="__BASE_URI__pictures/button1.png" />&nbsp;and then <img width="53" height="78" alt="" src="__BASE_URI__pictures/button2.png" />&nbsp;to enter the integral)</strong></p><p>&nbsp;</p><p>(b) Now find the area.</p><p>&nbsp;</p><p><2><span> </span></p><p>&nbsp;</p>@

qu.3.2.mode=Inline@
qu.3.2.name=BasicArea(x^2-a^2)@
qu.3.2.comment=<p>Since $F is below the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>-axis, we must integrate $F1 to obtain the area.</p>
<p>Area = $feedback<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$fb</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$fa</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ANS</mi></mrow></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=range(1,5);
$f=maple("x^2-($a)^2");
$F=mathml("y=$f");
$F1=mathml("-($f)");
$i="'int(-($f),x=-($a)..($a))'";
$M=maple("
fb:=eval(-x^3/3+($a)^2*x,x=$a):
fa:=eval(-x^3/3+($a)^2*x,x=-$a):
ans:=fb-fa:
convert(fb,string),
convert(fa,string),
convert(ans,string),
MathML[ExportPresentation]($i = -x^3/3 +($a)^2*x)");
$fb=switch(0,$M);
$fa=switch(1,$M);
$ANS=switch(2,$M);
$feedback=switch(3,$M);
$plot=plotmaple("plot($f,x=-($a)-1..($a)+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.3.2.uid=b673510c-925d-4701-9a32-6774a23da872@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Definite Integral;
  Sub-Topic=Area;
@
qu.3.2.weighting=1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.maple_answer=$i@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.libname=@
qu.3.2.part.1.mode=Maple@
qu.3.2.part.1.allow2d=2@
qu.3.2.part.1.plot=@
qu.3.2.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.1.type=maple@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.maple_answer=$ANS@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.libname=@
qu.3.2.part.2.mode=Maple@
qu.3.2.part.2.allow2d=1@
qu.3.2.part.2.plot=@
qu.3.2.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.2.type=formula@
qu.3.2.question=<p>Below is the graph of $F.</p><p>$plot</p><p>(a) The integral which gives the area bounded by $F and the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>axis from <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math> is</p><p><1></p><p>(<strong>In the equation editor, select <img width="54" height="41" src="__BASE_URI__pictures/button1.png" alt="" />&nbsp;and then <img width="53" height="78" src="__BASE_URI__pictures/button2.png" alt="" />&nbsp;to enter the integral)</strong></p><p>&nbsp;</p><p>(b) Now find the area.</p><p><2><span> </span></p><p>&nbsp;</p>@

qu.4.topic=AreaBetCurves@

qu.4.1.mode=Inline@
qu.4.1.name=LineMParab@
qu.4.1.comment=<p>Here the line is above the parabola, so we must integrat line minus parabola.</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=range(1,3);
$b=range(1,3);
$f="x^2+($a)*x";
$g="($b)*x+($b)*($a)";
$F=mathml("y=$f");
$G=mathml("y=$g");
$i="'int(($g)-($f),x=-($a)..($b))'";
$plot=plotmaple("plot([$f,$g],x=-($a)-1..($b)+1,color=[black,black],
thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$yr=($b)^2+($a)*($b);@
qu.4.1.uid=1b769610-6eb3-4017-8f42-2a079c66ff21@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Definite Integral;
  Sub-Topic=Area between curves;
@
qu.4.1.weighting=1,1,1,1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.answer=(-$a,0)@
qu.4.1.part.1.mode=Ntuple@
qu.4.1.part.2.editing=useHTML@
qu.4.1.part.2.question=(Unset)@
qu.4.1.part.2.name=sro_id_2@
qu.4.1.part.2.answer=($b,$yr)@
qu.4.1.part.2.mode=Ntuple@
qu.4.1.part.3.name=sro_id_3@
qu.4.1.part.3.maple_answer=$i@
qu.4.1.part.3.editing=useHTML@
qu.4.1.part.3.question=(Unset)@
qu.4.1.part.3.libname=@
qu.4.1.part.3.mode=Maple@
qu.4.1.part.3.allow2d=2@
qu.4.1.part.3.plot=@
qu.4.1.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.1.part.3.type=maple@
qu.4.1.part.4.name=sro_id_4@
qu.4.1.part.4.maple_answer=int(($g)-($f),x=-($a)..($b))@
qu.4.1.part.4.editing=useHTML@
qu.4.1.part.4.question=(Unset)@
qu.4.1.part.4.libname=@
qu.4.1.part.4.mode=Maple@
qu.4.1.part.4.allow2d=1@
qu.4.1.part.4.plot=@
qu.4.1.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.1.part.4.type=formula@
qu.4.1.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) Find the intersection points of $F and $G, entering the point with the smaller <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>value first.</p><p><1><span>, <2><span> </span></span></p><p>(b) The integral which gives the area bounded by $F and $G is</p><p><3><span> </span></p><p>(<strong>In the equation editor, select <img width="54" height="41" alt="" src="__BASE_URI__pictures/button1.png" />&nbsp;and then <img width="53" height="78" alt="" src="__BASE_URI__pictures/button2.png" />&nbsp;to enter the integral)</strong></p><p>(c) Now find the area.</p><p><4><span> </span></p><p>&nbsp;</p>@

qu.4.2.mode=Inline@
qu.4.2.name=ParabMLine@
qu.4.2.comment=<p>Here the parabola is above the line, so we must integrate the parabola minus the line.</p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=range(1,3);
$b=range(-3,-1);
$f="($a)*x-x^2";
$g="-($b)*x+($b)*($a)";
$F=mathml("y=$f");
$G=mathml("y=$g");
$plot=plotmaple("plot([$f,$g],x=($b)-1..($a)+1,color=[black,black],
thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$i="'int(($f)-($g),x=($b)..($a))'";
$yl=-($b)^2+($b)*($a);@
qu.4.2.uid=649d86dd-caa3-4c06-bb74-95eabda0976a@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Definite Integral;
  Sub-Topic=Area between curves;
  Difficulty=Easy;
@
qu.4.2.weighting=1,1,1,1@
qu.4.2.numbering=alpha@
qu.4.2.part.1.editing=useHTML@
qu.4.2.part.1.question=(Unset)@
qu.4.2.part.1.name=sro_id_1@
qu.4.2.part.1.answer=($b,$yl)@
qu.4.2.part.1.mode=Ntuple@
qu.4.2.part.2.editing=useHTML@
qu.4.2.part.2.question=(Unset)@
qu.4.2.part.2.name=sro_id_2@
qu.4.2.part.2.answer=($a,0)@
qu.4.2.part.2.mode=Ntuple@
qu.4.2.part.3.name=sro_id_3@
qu.4.2.part.3.maple_answer=$i@
qu.4.2.part.3.editing=useHTML@
qu.4.2.part.3.question=(Unset)@
qu.4.2.part.3.libname=@
qu.4.2.part.3.mode=Maple@
qu.4.2.part.3.allow2d=2@
qu.4.2.part.3.plot=@
qu.4.2.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.2.part.3.type=maple@
qu.4.2.part.4.name=sro_id_4@
qu.4.2.part.4.maple_answer=int(($f)-($g),x=($b)..($a))@
qu.4.2.part.4.editing=useHTML@
qu.4.2.part.4.question=(Unset)@
qu.4.2.part.4.libname=@
qu.4.2.part.4.mode=Maple@
qu.4.2.part.4.allow2d=1@
qu.4.2.part.4.plot=@
qu.4.2.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.2.part.4.type=formula@
qu.4.2.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) Find the intersection points of $F and $G, entering the point with the smaller <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>value first.</p><p><1><span>, <2><span> </span></span></p><p>&nbsp;</p><p>(b) The integral which gives the area bounded by $F and $G is</p><p><3><span> </span></p><p><strong>(In the equation editor, select <img width="54" height="41" alt="" src="__BASE_URI__pictures/button1.png" />&nbsp;and then <img width="53" height="78" alt="" src="__BASE_URI__pictures/button2.png" />&nbsp;to enter the integral.)</strong></p><p>&nbsp;</p><p>(c) Now find the area.</p><p><4><span> </span></p><p>&nbsp;</p>@

qu.4.3.mode=Inline@
qu.4.3.name=(a^2-x^2)-(x^2-a^2)@
qu.4.3.comment=<p>Here $G is above $F, so we must itegrate $step.</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=range(1,4);
$as=$a*$a;
$f="x^2-($as)";
$g="($as)-x^2";
$F=mathml("y=$f");
$G=mathml("y=$g");
$step=mathml("($g)-($f)");
$i="'int(($g)-($f),x=-($a)..($a))'";
$plot=plotmaple("plot([$f,$g],x=-($a)-1..($a)+1,color=[black,black],
thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.3.uid=73d12eb9-3717-4e98-b3f2-bdcd4ede4986@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
  Topic=Definite Integrals;
  Sub-Topic=Area between curves;
@
qu.4.3.weighting=1,1@
qu.4.3.numbering=alpha@
qu.4.3.part.1.name=sro_id_1@
qu.4.3.part.1.maple_answer=$i@
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.question=(Unset)@
qu.4.3.part.1.libname=@
qu.4.3.part.1.mode=Maple@
qu.4.3.part.1.allow2d=2@
qu.4.3.part.1.plot=@
qu.4.3.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.3.part.1.type=maple@
qu.4.3.part.2.name=sro_id_2@
qu.4.3.part.2.maple_answer=int(($g)-($f),x=-($a)..($a))@
qu.4.3.part.2.editing=useHTML@
qu.4.3.part.2.question=(Unset)@
qu.4.3.part.2.libname=@
qu.4.3.part.2.mode=Maple@
qu.4.3.part.2.allow2d=1@
qu.4.3.part.2.plot=@
qu.4.3.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.4.3.part.2.type=formula@
qu.4.3.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) The integral which gives the area bounded by $F and $G is</p><p><1><span> </span></p><p><strong>(In the equation editor, select <img width="54" height="41" src="__BASE_URI__pictures/button1.png" alt="" />&nbsp;and then <img width="53" height="78" src="__BASE_URI__pictures/button2.png" alt="" />&nbsp;to enter the integral).</strong></p><p>(b) Now find the area.</p><p><2><span> </span></p><p>&nbsp;</p>@

qu.5.topic=two integrals@

qu.5.1.mode=Inline@
qu.5.1.name=TwoIntegrals@
qu.5.1.comment=<p>From <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math>$G is above $F, meaning we must integrate $step1.</p>
<p>From <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></math>&nbsp;to <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></math>$F is above $G, meaning we must itegrate $step2.</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$z=rint(3);
$f=switch($z,"x^(1/3)","sin(Pi/2*x)","x^3");
$f1=switch($z,"surd(x,3)","sin(Pi/2*x)","x^3");
$g=switch($z,"0","x","0");
$F=mathml("y=$f");
$G=mathml("y=$g");
$iL="'int(($g)-($f),x=-1..0)'";
$iR="'int(($f)-($g),x=0..1)'";
$plot=plotmaple("plot([[x,$f1,x=-1.1..1.1],[x,$g,x=-1.1..1.1],
[-1,x,x=-1..0],[1,x,x=0..1]],color=[black, black,red,red],
thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");

$step1=mathml("($g)-($f)");
$step2=mathml("($f)-($g)");@
qu.5.1.uid=e4740d08-cb3e-4b7d-bfa2-2e65e616a853@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Definite Integrals;
  Sub-Topic=Area between curves, two integrals;
@
qu.5.1.weighting=1,1,1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.maple_answer=$iL@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.question=(Unset)@
qu.5.1.part.1.libname=@
qu.5.1.part.1.mode=Maple@
qu.5.1.part.1.allow2d=2@
qu.5.1.part.1.plot=@
qu.5.1.part.1.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.5.1.part.1.type=maple@
qu.5.1.part.2.name=sro_id_2@
qu.5.1.part.2.maple_answer=$iR@
qu.5.1.part.2.editing=useHTML@
qu.5.1.part.2.question=(Unset)@
qu.5.1.part.2.libname=@
qu.5.1.part.2.mode=Maple@
qu.5.1.part.2.allow2d=2@
qu.5.1.part.2.plot=@
qu.5.1.part.2.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.5.1.part.2.type=maple@
qu.5.1.part.3.name=sro_id_3@
qu.5.1.part.3.maple_answer=int(($g)-($f1),x=-1..0)+int($f1-$g,x=0..1)@
qu.5.1.part.3.editing=useHTML@
qu.5.1.part.3.question=(Unset)@
qu.5.1.part.3.libname=@
qu.5.1.part.3.mode=Maple@
qu.5.1.part.3.allow2d=1@
qu.5.1.part.3.plot=@
qu.5.1.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.5.1.part.3.type=formula@
qu.5.1.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) The area bounded by $F and $G is given by the sum of these two integrals. (Enter the integral with the smaller bounds of integration first.)</p><p><1><span> </span></p><p><span><2><span> </span></span></p><p>(<strong>In the equation editor, select <img alt="" width="54" height="41" src="__BASE_URI__pictures/button1.png" />&nbsp;and then <img alt="" width="53" height="78" src="__BASE_URI__pictures/button2.png" />&nbsp;to enter the integral)</strong></p><p>(b) Now find the area.</p><p><3><span> </span></p><p>&nbsp;</p>@

qu.6.topic=wrty@

qu.6.1.mode=Inline@
qu.6.1.name=ParabMLineY@
qu.6.1.comment=<p>Did you flip the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values when finding the intersection points? Remember, here you are finding the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values first and then substituting to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>.</p>
<p>Here the parabola has the larger <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>values, meaning we must integrate parabola minus line: $step.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=range(1,3);
$b=range(-3,-1);
$f="($a)*y-y^2";
$g="-($b)*y+($b)*($a)";
$F=mathml("x=$f");
$G=mathml("x=$g");
$step=mathml("($f)-($g)");
$plot=plotmaple("plot([[$f,y,y=($b)-1..($a)+1],[$g,y,y=($b)-1..($a)+1]],color=[black,black],
thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$i="'int(($f)-($g),y=($b)..($a))'";
$xl=-($b)^2+($b)*($a);@
qu.6.1.uid=99da5a52-f50f-4240-863c-3290dd262058@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Definite Integrals;
  Sub-Topic=Area between curves, wrt to y;
@
qu.6.1.weighting=1,1,1,1@
qu.6.1.numbering=alpha@
qu.6.1.part.1.editing=useHTML@
qu.6.1.part.1.question=(Unset)@
qu.6.1.part.1.name=sro_id_1@
qu.6.1.part.1.answer=($xl,$b)@
qu.6.1.part.1.mode=Ntuple@
qu.6.1.part.2.editing=useHTML@
qu.6.1.part.2.question=(Unset)@
qu.6.1.part.2.name=sro_id_2@
qu.6.1.part.2.answer=(0,$a)@
qu.6.1.part.2.mode=Ntuple@
qu.6.1.part.3.name=sro_id_3@
qu.6.1.part.3.maple_answer=$i@
qu.6.1.part.3.editing=useHTML@
qu.6.1.part.3.question=(Unset)@
qu.6.1.part.3.libname=@
qu.6.1.part.3.mode=Maple@
qu.6.1.part.3.allow2d=2@
qu.6.1.part.3.plot=@
qu.6.1.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.6.1.part.3.type=maple@
qu.6.1.part.4.name=sro_id_4@
qu.6.1.part.4.maple_answer=int(($f)-($g),y=($b)..($a))@
qu.6.1.part.4.editing=useHTML@
qu.6.1.part.4.question=(Unset)@
qu.6.1.part.4.libname=@
qu.6.1.part.4.mode=Maple@
qu.6.1.part.4.allow2d=1@
qu.6.1.part.4.plot=@
qu.6.1.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.6.1.part.4.type=formula@
qu.6.1.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) Find the intersection points of $F and $G, entering the point with the smaller <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>value first.</p><p>&nbsp;</p><p><1><span>, <2></span></p><p><span><span> </span></span></p><p>(b) The integral in terms of<strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math></strong>which gives the area bounded by $F and $G is</p><p>&nbsp;</p><p><3><span> </span></p><p>(<strong>In the equation editor, select <img width="54" height="41" src="__BASE_URI__pictures/button1.png" alt="" />&nbsp;and then <img width="53" height="78" src="__BASE_URI__pictures/button2.png" alt="" />&nbsp;to enter the integral)</strong></p><p>&nbsp;</p><p>(c) Now find the area.</p><p>&nbsp;</p><p><4><span> </span></p><p>&nbsp;</p>@

qu.6.2.mode=Inline@
qu.6.2.name=LineMParabY@
qu.6.2.comment=<p>Did you flip the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values when finding the intersection points? Remember, here you are finding the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>values first and then substituting to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>.</p>
<p>Here the line has the larger <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>values, so we must integrate line minus parabola: $step.</p>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=range(1,3);
$b=range(1,3);
$f="y^2+($a)*y";
$g="($b)*y+($b)*($a)";
$F=mathml("x=$f");
$G=mathml("x=$g");
$step=mathml("($g)-($f)");
$i="'int(($g)-($f),y=-($a)..($b))'";
$plot=plotmaple("plot([[$f,y,y=-($a)-1..($b)+1],[$g,y,y=-($a)-1..($b)+1]],
color=[black,black],thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$xl=($b)^2+($a)*($b);@
qu.6.2.uid=a5baa150-c02d-418a-9cd8-24a34d926709@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Topic=Definite Integrals;
  Sub-Topic=Area between curves, wrt y;
  Course=Introduction to Calculus I;
@
qu.6.2.weighting=1,1,1,1@
qu.6.2.numbering=alpha@
qu.6.2.part.1.editing=useHTML@
qu.6.2.part.1.question=(Unset)@
qu.6.2.part.1.name=sro_id_1@
qu.6.2.part.1.answer=(0,-$a)@
qu.6.2.part.1.mode=Ntuple@
qu.6.2.part.2.editing=useHTML@
qu.6.2.part.2.question=(Unset)@
qu.6.2.part.2.name=sro_id_2@
qu.6.2.part.2.answer=($xl,$b)@
qu.6.2.part.2.mode=Ntuple@
qu.6.2.part.3.name=sro_id_3@
qu.6.2.part.3.maple_answer=$i@
qu.6.2.part.3.editing=useHTML@
qu.6.2.part.3.question=(Unset)@
qu.6.2.part.3.libname=@
qu.6.2.part.3.mode=Maple@
qu.6.2.part.3.allow2d=2@
qu.6.2.part.3.plot=@
qu.6.2.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.6.2.part.3.type=maple@
qu.6.2.part.4.name=sro_id_4@
qu.6.2.part.4.maple_answer=int(($g)-($f),y=-($a)..($b))@
qu.6.2.part.4.editing=useHTML@
qu.6.2.part.4.question=(Unset)@
qu.6.2.part.4.libname=@
qu.6.2.part.4.mode=Maple@
qu.6.2.part.4.allow2d=1@
qu.6.2.part.4.plot=@
qu.6.2.part.4.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.6.2.part.4.type=formula@
qu.6.2.question=<p>Below is the graph of $F and $G.</p><p>$plot</p><p>(a) Find the intersection points of $F and $G, entering the point with the smaller <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math>value first.</p><p>&nbsp;</p><p><1><span>, <2></span></p><p><span><span> </span></span></p><p>(b) The integral in terms of<strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi></mrow></math></strong>which gives the area bounded by $F and $G is</p><p>&nbsp;</p><p><3><span> </span></p><p><strong>(In the equation editor, select <img width="54" height="41" src="__BASE_URI__pictures/button1.png" alt="" />&nbsp;and then <img width="53" height="78" src="__BASE_URI__pictures/button2.png" alt="" />&nbsp;to enter the integral)</strong></p><p>&nbsp;</p><p>(c) Now find the area.</p><p>&nbsp;</p><p><4><span> </span></p><p>&nbsp;</p>@

