Question:

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

My answers:

(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.

(b)

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: