# Difficult Region of Integration Involving Gauss's Theorem

I'm told to use Gauss's Theorem to compute the flux of a field $$\vec F = $$ along the boundary of the cylindrical solid $$x^2+y^2 \le 4$$ below $$z=8$$ and above $$z=x$$.

I know by Gauss's Theorem that:

Net Flux = $$\iint_{\partial D} \vec F \cdot \vec ndS = \iiint_D \nabla \cdot \vec FdV$$

This computation is pretty straight forward. $$\nabla \cdot \vec F = 2+2y$$. But the region of integration is particularly difficult to map out.

I thought to use cylindrical coordinates and setting the bounds to $$0 \le \theta \le 2 \pi$$, $$0 \le z \le 8$$, and $$0 \le r \le 4$$, but this seems like it would just give me the area of the cylinder of height 8--and wouldn't include the part where z=x slices through the cylinder.

What would be the right way to go in terms of the bounds of integration?

You have the $$\theta$$ and $$r$$ bounds right, as well as the upper bound for $$z$$. The lower bound can be thought of as the lower "boundary" of your region, i.e., $$z = x$$ or $$z = r \cos \theta$$ in cylindrical coordinates. So your bounds would be $$r\cos \theta \leq z \leq 8$$