Verify Green's Theorem-Calculate $\int \int_R{ \nabla \times \overrightarrow{F} \cdot \hat{n}}dA$

Question

Given that $$\vec{F}=-x^2y \hat{i}+x y^2\hat{j}$$ $$C:r=a \cos{t}\hat{\imath}+a \sin{t} \hat{\jmath}, 0 \leq t \leq 2 \pi \text{ and } R: x^2+y^2 \leq a^2$$ I have to calculate $\iint_R{ \nabla \times \vec{F} \cdot \hat{n}}\,dA$. $$$$ $$\nabla \times \vec{F}=(x^2+y^2)\hat{k}$$ $$\hat{n}=\hat{k}$$ So $$\iint_R{ \nabla \times \vec{F} \cdot \hat{n}}\,dA=\iint_R{ \nabla \times \vec{F} \cdot \hat{k}}\,dA=\iint_R{x^2+y^2}dA$$ But how can I continue?? Do I have to do something like the following?? $$\iint_R{x^2+y^2}\,dA=\int_{-1}^1 \int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}{(x^2+y^2)}\,dy\,dx$$

Answer 1

Make the change of variable $(x,y)=(r\cos t, r\sin t)$ into the integral, using $dA = dxdy = rdrdt$:

$$ \int\int_R (x^2 + y^2) dA= {2\pi}\int_0^{a} r^2 rdr = {2\pi}\frac{a^4}4 $$