The question I am working on gives the Poisson Equation for Newtonian Gravity $∇⃗ ∙ ⃗(⃗) = −4(⃗) $ with the mass density function $(⃗) =0/(1 + (⁄0)^2)$ and asks for the flux of the gravitational acceleration across a spherical surface with radius $r$.
Using the divergence theorem, so far I have:
$ \oint \oint \overrightarrow{g} (\overrightarrow{r}) \cdot ds = \iiint \overrightarrow{\nabla} \cdot \overrightarrow{g} (\overrightarrow{r}) dV$
$= -4\pi G \rho 0 \iiint 1/(1+(r/r0)^2) dV$
However I don't know how to solve the integral I have arrived at. I thought about changing it to spherical coordinates but am not sure if that would help?
"Slicing" along the z-axis yields circles of radius $\sqrt{r^2-z^2}$, so parameterizing in terms of $r$ and $\theta$: $$ \int_0^{\sqrt{r^2-z^2}}\int_0^{2\pi}\frac{r}{1+(\frac{r}{r_0})^2}d\theta dr $$ So the entire integral should be: $$ \iiint\nabla\cdot\vec{g}(\vec{r})dV=-4\pi G\rho_0\int_{-r}^r\int_0^{\sqrt{r^2-z^2}}\int_0^{2\pi}\frac{r}{1+(\frac{r}{r_0})^2}d\theta dr dz $$ Which can be solved with integration by parts and partial fractions.