I'm trying to solve the integral of the following function in a sphere of radius $5\sigma$
the function is: $$f(r) = \frac{r^{2}}{(2 \pi \sigma^{2})^{\frac{3}{2}}}\exp\left(-\frac{1}{2}(\frac{r}{\sigma})^{2}\right)$$
and i have to compute the following integral $2\pi^{2}\int_{0}^{5\sigma}f(r)dr$ (the factor of $2 \pi^{2}$ comes from the integration along $\theta$ and $\phi$ in spherical coordinates)
by solving with mathematematica i get that the result is independent of $\sigma$ in particular i get 1.57077
Moreover ,whatever $\sigma$ i choose, the integral result tends to $\frac{\pi}{2}$ as the upper limit of the integral goes to $\infty$ (With a fixed $\sigma$)
is this correct?
Another remark: i'm also solving this integral with a finite element method (gaussian quadrature) on a cubic mesh. Also using this method the integral is independent from $\sigma$ however as i raise the upper limit of the integral, in this case, the integral result asymptotize to $1$. There's a ratio between the results that is exactly $\frac{\pi}{2}$. Any idea of some possible mistake?