Evaluate $\oint_C -x^2 y \ dx + xy^2 \ dy$ using Green's Theorem

Question

I am asked to find

$$\oint_C -x^2 y \ dx + xy^2 \ dy$$

using Green's Theorem, where C is the circumference with radius $2$ and center on the origin. My question is, should I use the Jacobian for this?

My answer

\begin{align*} &\frac{\partial Q}{\partial x} = y^2 \quad \frac{\partial P}{\partial y} = -x^2\\ & \oint_C -x^2 y \ dx + xy^2 \ dy = \iint_D x^2 + y^2 \ dA = \int_{0}^{2\pi} \int_{0}^{2} r^2 \ r \ drd\theta = \cdots = 8\pi \end{align*}

Answer 1

Yes, it correct. The same result turns out after the direct evaluation \begin{align*} \oint_C -x^2 y \ dx + xy^2 \ dy&=\int_0^{2\pi}( -x^2 y x' + xy^2 y')dt\\ &=32\int_0^{2\pi} \sin^2(t)\cos^2(t) dt=4\int_0^{2\pi} 2\sin^2(2t) dt\\ &=4\int_0^{2\pi} (\sin^2(2t)+\cos^2(2t)) dt=8\pi \end{align*} where $x(t)=2\cos(t)$ and $y(t)=2\sin(t)$ with $t\in [0,2\pi]$.