1. ## Green's Theorem

I'm just trying to learn Green's Theorem. Can somebody look this over and see if I've got it right?

Question:

Let R be a region bounded by the curve and line

$y=-x^2+3x-2$ and $y=-2$

Verify Green's theorem in the plane for

$\oint_{\partial{R}} (x-y)dx (x-y)dy$

For the region I found $0 \le x\le 3$ and $-2 \le y \le 1/4$

So:

$\int_0^3 \int_{-2}^{\frac{1}{4}} \frac{\partial}{\partial x}(x-y)-\frac{\partial}{\partial y}(x-y)dydx$

$=\int_0^3 \int_{-2}^{\frac{1}{4}}2dydx$

$=\int_0^3 \frac{9}{2} dx = \frac{27}{2}$

2. The integrand is correct but the limits are not--the region is not a rectangle

See attachment

You can also verify by computing the line integral directly though in this case it would be some work