Volume with triple integrals Calculate integral $$\iiint_V \frac{e^{-x^2-y^2-z^2}}{\sqrt{x^2+y^2+z^2}} dV$$ Where $V\subset\mathbb{R}^3$ is the exterior of a origocentered sphere with radius of 2 
\begin{align*}
V=&\iiint_V \frac{e^{-x^2-y^2-z^2}}{\sqrt{x^2+y^2+z^2}} dV \\
=& \int_0^{2\pi}\int_0^{\pi}\int_0^2 \frac{e^{-r^2}}{r}\rho^2\sin\phi dr d\phi d \theta \\
=& 2\pi \Biggl[-\cos\phi\Biggr]_0^{\pi} \int_0^2 re^{-r^2} dr \\
=& 2\pi (\underbrace{-\cos\pi}_{=1} +\underbrace{\cos0}_{=1}) \int_0^2 re^{-r^2} dr \\
=& 4\pi  \Biggl[\frac{-e^{-r^2}}{2}\Biggr]_0^2 \\
=& 2\pi  \left( -e^{-2^2}+1\right) \\
=& 2\pi -2\pi e^{-4}
\end{align*}

Converting to polar coordinates yields
$$
\begin{align}
\int_0^2\frac{e^{-r^2}}{r}4\pi r^2\,\mathrm{d}r
&=2\pi\int_0^4 e^{-t}\,\mathrm{d}t\\
\end{align}
$$
using the substitution $t=r^2$
