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Funny behavior with int

Alex Smith 630
December 01 2007
1
My Calculus III students stumbled on this buggy thing while evaluating a line integral to calculate the flux The curve [X(t),Y(t)] is the right-half of a Lemniscate with polar equation R^2=cos(2*theta). The vector field is F(x,y)=M(x,y)i+N(x,y)j. They were integrating M*dy-N*dx around the curve. If we let a=M*dy and aa=expand(M*dy), then they find that Maple's int gives inconsistent results. As far as I can tell, a and its twin aa are well-behaved over -Pi/4..Pi/4 and equal. Maybe it is a bug in how Maple handles elliptic integrals? Or maybe it is some issue with removable discontinuities? Code follows: notice how int(a,t=-Pi/4..Pi/4) and int(aa,t=-Pi/4..Pi/4) are not equal, but ought to be equal. ........ X:=t->cos(t)*sqrt(cos(2*t)); Y:=t->sin(t)*sqrt(cos(2*t)); M:=(x,y)->x^2+4*y; N:=(x,y)->x+y^2; a:=M(X(t),Y(t))*diff(Y(t),t); aa:=expand(a); int(a,t=-Pi/4..Pi/4); int(aa,t=-Pi/4..Pi/4); Int(aa,t=-Pi/4..Pi/4);evalf(%); plot([a,aa+.1],t=-Pi/4..Pi/4);
plot integral

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