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I'm told to use Gauss's Theorem to compute the flux of a field $\vec F = <x,y^2,y+z>$ 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?

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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 $$r\cos \theta \leq z \leq 8$$

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