# Thread: Green's Theorem Question

1. ## Green's Theorem Question

I'm trying to evaluate the line integral $F(x,y)=\left ({\arctan y,\frac{-xy^{2}}{1+y^{2}}} \right )$ around the curve $4x^{2}+9y^{2}=36$

Does anybody else get -6Pi?

2. I just punched the integral into Mathematica and also received $-6\pi$.

3. By Green's theorem

$\oint \limits_C Pdx + Qdy = \iint \limits_R Q_x - P_y dA$

which for your particular vector field gives

$\iint \limits_R -1 dA$ and the area of the ellipse is $6 \pi$, hence your answer of $- 6 \pi$.