let $C$ be a closed $C^1$path (oriented counterclock wise) consisting of a piece of $y^2=2(x+2)$ and the vertical segment $x=2$, and $F$ is a vector field such that $$F(x,y)=\left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right)=(P,Q)$$ evaluate the integral along $C$ using green's theorm.
let $P$ be the region bounded by $C$, then the double integral will look like this $$I=\oint_CF=\int_{-2}^2\int_{-\sqrt{2x+4}}^{\sqrt{2x+4}}\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\,dy\,dx$$ I have two problems the first one is that the bounds are quite hard so I have to change variables but I don't know which map is the best, and the second one is that the integrand is equal to $0$ which means that $I$ is $0$ as well, but my textbook says that it's $8$.
Well, the answer is not $8$. Your textbook should show you an example like this. Draw a circle centered at the origin that lies inside your curve. Now use Green’s Theorem on the region between the two curves.