Let $E$ be defined in $\mathbb{R}^3$ as the area between the parabola $z = x^2 + y^2$ and the plane $z = 2$. Given the vector $\vec{F} = \langle 2x, 2y, 0 \rangle$, find the flux integral $\iint_S \vec{F}\cdot\mathrm{d}\vec{r}$ using the divergence theorem.
So, we are looking for $$\iiint_E \text{div}\,\vec{F}\,\mathrm{d}V$$ Finding the divergence of $\vec{F}$ is easy, but I've been having trouble with the bounds. How do you set up the bounds for this integral?
$$ V=\int_0^2 \int _{-\sqrt z} ^{\sqrt z} \int_{-\sqrt{z-y^2}}^{\sqrt{z-y^2}}dxdy dz $$
$$ V=2\int_0^2 \int _{-\sqrt z} ^{\sqrt z} \sqrt{z-y^2} dy dz $$ let $y =\sqrt z \sin \theta$ so that $$.$$ $dy = \sqrt z \cos\theta d \theta$ $$.$$and $\sqrt{z-y^2} = \sqrt z \cos \theta$
$$ V=2\int_0^2 z\int _ {-\frac \pi 2} ^{\frac \pi 2} \cos ^2 \theta d\theta dz $$ $$ \implies V = \pi \int_0^2 z\; dz = 2 \pi$$
Which is the same answer you get by integrating cross sections parallel to the $x-y$ plane, which are circles having radius $r = \sqrt z$ so each infinitesimal cylinder has volume $dV = \pi z dz$