Question: Let $C$ be closed curve as the boundary of region $R$.
$C$ is defined as the polar coordinate inequalities $1\le r\le2, 0\le t\le \pi$.
Define the field $F(x,y)=P(x,y)i+Q(x,y)j$ where $P=x^2+y^2,Q=y^3+x^2$. Using Green's theorem, find the counterclockwise circulation.
My attempt:
$ \oint_C Pdx+Qdy =\iint_R (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}) \,dA=\iint_R (2x-2y) \,dA$.
Now my idea is to convert $x,y$ coordinates into polar coordinates and we then get:
$\iint_R (2x-2y) \,dA=\int_0^{\pi}\int_1^2(2r\cos(t)-2r\sin(t))rdrdt$.
Is my idea right/valid here? If not, how should I approach this problem?