
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
$\displaystyle y=x^2+3x2$ and $\displaystyle y=2$
Verify Green's theorem in the plane for
$\displaystyle \oint_{\partial{R}} (xy)dx (xy)dy$
For the region I found $\displaystyle 0 \le x\le 3$ and $\displaystyle 2 \le y \le 1/4$
So:
$\displaystyle \int_0^3 \int_{2}^{\frac{1}{4}} \frac{\partial}{\partial x}(xy)\frac{\partial}{\partial y}(xy)dydx$
$\displaystyle =\int_0^3 \int_{2}^{\frac{1}{4}}2dydx$
$\displaystyle =\int_0^3 \frac{9}{2} dx = \frac{27}{2}$

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The integrand is correct but the limits are notthe 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