Compute $$ \oint_{C}\left[\left(2x - y^{3}\right)\mathrm{d}x - xy\,\mathrm{d}y\right], $$ where $C$ is the boundary of region enclosed by $x^{2} + y^{2} = 1$ and $x^{2} + y^{2} = 9$.
I am confused because it's integration is under closed sign but both the curves are different which don't intersect then what should by my limit ?.
$$\oint_C (2x-y^3)dx-(xy)dy=\iint_R\left(\frac{\partial(-xy)}{\partial x}-\left(\frac{\partial(2x-y^3)}{\partial y}\right)\right)\,dA=$$
$$=\iint_R\left(-y-3y^2\right)\,dx\,dy=\int_0^{2\pi}\int_1^3 r(-r\sin\theta-3r^2\sin^2\theta)\,dr\,d\theta=$$
$$=\int_0^{2\pi}\int_1^3-\left(r^2\sin\theta+3r^3\sin^2\theta\right)=\int_0^{2\pi}\left(-\frac{26}3\sin\theta-60\sin^2\theta\right)\,d\theta=$$
$$=0-\left.30(\theta-\sin\theta\cos\theta)\right|_0^{2\pi}=-60\pi$$
$$=$$