I have a function $f(r)$ where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\forall x,y,z \geq 0$. I know the values of the function at many points, essentially I have a table of values with $r$ and the corresponding value $f(r)$, let's call these $\textbf{data}$. I am trying to calculate the integral
$\int dz'~ e^{-\frac{(z-z')}{2\sigma^2}} f(\sqrt{x^{2}+y^{2}},z')$ over the region of interest, where $\sigma$ is a parameter whose value is known.
Here is what I did:
I uniformly picked $n$ samples of $\sqrt{x^{2} + y^{2}}$ and $z$ in the region of interest, where $n$ is the number of data points. For each of value of $\sqrt{x^{2} + y^{2}}$, I calculated $z'=\sqrt{r^{2}- x^{2} + y^{2}}$, where $r$ is from the data. I evaluated the integrand for the list of these values. I then used $\textbf{Simpson's rule}$ to evaluate the integral over the range of $z'$.
Is this a valid method to evaluate this integral?
Thanks