I'm stuck at an exercise with a surface integral $\int_Y yz\,\mathrm dS$, where the surface $Y$ is given by the sphere $4=x^2+y^2+z^2$, above the region $x\leq 0, -x\leq y, 0\leq x^2+y^2\leq 4$.
I started off with doing polar coordinates. And that's as far as I've come the last hours. Can somebody help me in the right direction?
The surface $Y$ is a part of the sphere that has $z$ symmetry i.e. $$(x,y,z)\in Y \implies (x,y,-z) \in Y$$ The integrand is an odd function of $z$. Therefore we conclude that the integral is $0$ by symmetry.