Solving surface integral

Question

I just need some help on solving surface integral. Actually I already finished doing part (a) and part (b) but just part (c) I dont know how to do it. It would be nice if someone able to guide me to set up the problem.

For part (a) I got my answer $5/3$ $\pi$

and part (b) I got $2/3$ $\pi$

Thanks

Answer 1

From the divergence theorem you have: $$\int\int_{S_1} f\cdot n_1\,dS+ \int\int_{S_2} f\cdot n_2 \,dS= \int\int\int_D \nabla \cdot f \,dV,$$ where $\partial V=S_1\cup S_2$, $n_1$ and $n_2$ are the exterior normal unit vector of $S_1$ and $S_2$.