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

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

Verify Green's theorem in the plane for

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

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

So:

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

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

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