Green's theorem question needs checking
Use Green's theorem to evaluate the following line integrals. Assume that the curve is transversed counterclockwise.
(a) where is the rectangle with corners at (-2,1),(-2,2),(4,1) & (4,2).
(b) where is the circle
(a) P=3xy & Q=2xy
Green's Theorem gives:
where D is the rectangle enclosed by
I found if I just go ahead and integrate I end up with 0 as the answer.
However, since the path is not smooth, should I take the integral along each side separately and then add them?
Doing this I end up with:
first side (-2,1) to (-2,2). Make x=-2 -> dx = 0. Make y=t -> dy=dt.
I did the same for the other 3 sides and got 12, 18 and 36. This makes the entire integral 60.
Does this look ok? Or do I need to take the absolute values of each, making the final answer 72.
Green's theorem gives:
where D is a disk of radius 4 centred at the origin.
Since this is a disk, is the best way to do this integral is using polar co-ordinates?
This is what I got: