I am stuck on a few double integral questions.
1. Let D be the region in the xy plane bounded by the curves r = cos(theta) and r = sin(2theta)/sqrt(2). Compute the double integral :
In Polar coordinates this region is described by two sets:
A: theta from 0 to pi/4 r from 0 to (sin (2 theta)/ sqrt (2)
B: theta from pi/4 to pi/2 from 0 to cos theta
I tried the changed of variables formula, substituting x for r*cos (theta) and multiplying by the Jacobian r, but this does not give me a nice integration at all. Is there something I am doing wrong?
2. Let D the region in the xy plane enclosed by x+1 = 1, x+y = 3, x = 0 and y = 0. Evaluate the integral:
Finding an Inverse Mapping: I found that the Jacobian for the inverse mapping from (x,y) to (u,v) is F^-1 (u,v) = ((u+v)/2, (u-v)/2) = (x,y). This gives me 1/2 as the absolute value of the Jacobian. And a region bounded by the lines u=1,u=3, v=u, v=-u.
I then evaluate this integral using the image under F and the following bounds:
v between u and -u, and u between 1 and 3.
This of course does not give me a good integration. I think I am confused as to when I have to multiply by the Jacobian.
Any insight would be helpful, Thanks in advance.