Originally Posted by

**alexmahone** $\displaystyle \vec{v}=\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$

$\displaystyle div\ \vec{v}=\frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z}=3$

$\displaystyle V=\pi r^2h=\pi*2^2*(3-1)=8\pi$

Using the divergence theorem,

$\displaystyle \iint\limits_S \, \vec{v}(\vec{r})\,dS=\iiint\limits_V \, div\ \vec{v}\, dV=3*8\pi=24\pi$

(In the solution, the volume is erroneously calculated as $\displaystyle V=\pi*2^2*3=12\pi$ and so $\displaystyle \iint\limits_S \, \vec{v}(\vec{r})\,dS=3*12\pi=36\pi$