Let $C$ be the intersection curve of the parabolic sheet $y=x^2$ with the cylinder $x^2+z^2=4$, oriented clockwise when viewed from the positive $y$-axis. Apply Stoke's Theorem to the integral
$$ \int_C 2y\,dx+xz\,dy+z^2\,dz$$ and continue until you have an iterated double integral. Do not solve.
So far I computed the curl to be $(-x,0,z-2)$ , and what I attemped to do was plug $y=x^2$ into the cylinder equation and found the normal to be $(0,1,2z)$ and then found the double integral bounds to be $0<z<\sqrt{4-x^2}$ and $-2<x<2$. Am I on the right track so far? I have no idea if I'm proceeding on the right direction so any pointers would be appreciated.
My final integral that I end up with after plugging everything in is $$\int_2^2\int_0^\sqrt{4-x^2} \ 2z^2-4z\, dz\, dx.$$
