Use Green’s theorem to compute $\int_C x^2 ydx − xy^2 dy$ where $C$ is the circle $x^2 + y ^2 = 4$ oriented counter-clockwise.

Using Green's theorem we have
\[\int_C x^2ydx-xy^2dy=\iint_D \frac{\partial}{\partial x}(-xy^2)- \frac{\partial}{\partial y}(x^2y)dA=\iint_D-y^2-x^2dA\]
\[=\int_0^2\int_0^{2\pi}-r^2 (rd\theta dr)=-\int_0^{2}r^3dr \int_0^{2\pi}d\theta\]
\[=-\frac{r^4}{4}|_0^{2}\times 2 \pi=-4\pi,\]
Here $D$ is the circle of radius $2$ centered at the origion.