I'm trying to solve the following problem
Let $C$ be the curve in the $xy$ plane given in polar coordinates by $r = 2-\sin(\theta),\ 0 \leq \theta \leq \pi$ and let $S$ be the surface given by revolution of $C$ round the $x$-axis. Calculate the flux through $S$ of the following vector field: $$ F(x,y,z) = \bigl(y^2+z^2,\, \cos(z^3) - \ln(7+e^{2z^2}),\, 5x+z\bigr) $$
Now, I've noted the divergence of $F$ is equal to $1$, but couldn't find a way to calculate the volume of the solid bounded by $S$. I've also parametrized $S$ and tried calculating the flux of $F_2=(0,0,z)$ (which by the divergence theorem I believe should be equal to the flux of $F$) through $S$ directly but ended up with an unmanageable integral. I've kind of ran out of ideas so any help would be greatly appreciated.