Use The Divergence Theorem to evaluate the flux

Question

Let $ \vec{F}(x, y, z) = (\sin z + xy^2)\vec{i} + x^2e^{3z}\vec{j} + (\cos ^3x + x^2z)\vec{k}. $

Let $ T $ be the surface bounding the region $ R $ given by $ x^2 + y^2 \leq z \leq 6 - \sqrt{x^2 + y^2}, $ oriented outward. Use the Divergence Theorem to evaluate the flux: $$ \iint_{T} \vec{F}.d\vec{S} $$

I am having trouble applying the Divergence Theorem to calculate the flux, I got $$ \operatorname{div}\vec{F} = \left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z}\right) \vec F = y^2 + 0 + x^2 $$ but then I don't know how to proceed.

Answer 1

Switch to polar coordinates to do the triple integral:

$x=r\cos \theta ;\ x=r\sin \theta ;\ z=z$.

Then, applying the Divergence Theorem amounts to evaluating

$\int_{0}^{2\pi }\int_{0}^{6}\int_{r^{2}}^{6-r}r^{3}dzdrd\theta .$