I'm currently stuck with this kind of problem:
Let $S$ be the surface in $\mathbb{R}^3$ obtained by evolving the curve
$x = \cos u$ and $z = \sin 2u$ where $-\frac{\pi}{2} \leq u \leq \frac{\pi}{2}$
in the $xz$-plane around the $z$-axis. Compute the volume of the region bounded by $S$.
Here I try to Let $F=(0,0,z)$ thus $div\vec{F}=1$, then $$\iiint_E \ div\vec{F}\,dx\,dy\,dz$$ = $$\iiint_E 3\,dV$$ = 3Volume. However I'm not sure how to process the other side of the equation given Divergence Theorem, i.e $$\unicode{x222F}_S \vec{F}\cdot\vec{n}\,dS$$=?