Line integral applying greens theorem

Question

I want to calculate line integral:

$$\oint_Cy^2\,dx+3xy\,dy$$

when $C$ is half circle $D=(x,y):x^2+y^2\le1,y\ge0$

I know that $r=1$

The applying greens theorem

$$\frac{\delta Q}{\delta x}=3y$$

and

$$\frac{\delta P}{\delta y}=2y$$

so

$$3y-2y=y$$

then the integral

$$\int_0^1y\,dy=\frac{1}{2}$$

is this legit way to calculate this?

Answer 1

By Green's theorem, $$\oint_CP\,dx+Q\,dy={\int\int}_D\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\,dA$$ where $D$ is the region enclosed by the curve. As you mentioned $r=1$ but you also need the bounds for the angle $\theta$ which is $0$ to $\pi$ since you are looking at the upper half circle. Also $dA=r\,dr\,d\theta$ for the polar transformation and $y=r\sin\theta$. Therefore,

$$\oint_Cy^2\,dx+3xy\,dy=\int_{0}^{\pi}\int_{0}^{1}r\sin(\theta) r \,dr\,d\theta$$