Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (-2,1), (-2,-3), (1,-1), (1,5) and back to (-2,1), in that order. Use Green's Theorem to evaluate the integral ∫(2xy)dx+(xy2)dy.
∫Pdx+Qdy = ∫∫(∂Q/∂x-∂P/∂y)dxdy
The Attempt at a Solution
I first found ∂Q/∂x to be 2y2 and ∂P/∂y to be 2x.
I am struggling to find bounds that are easy to use for the double integral. I tried keeping x fixed and integrating from the y = (2/3)x-5/3 to y = (4/3)x+11/3. But that got messy quickly.
Then I tried keeping y fixed and integrating from x = (3/2)y+(5/2) and x=(3/4)y-11/4. I forced my way through that one, but there has got to be a simpler way. Furthermore, I got a wrong answer that way anyway, but I don't know if it's because I'm setting the integral up incorrectly or if I made an error.
Thanks and sorry if the formatting looks weird I've never used this site before.