I have a scalar field $f(x,y,z)$ defined over a finite region, and what's important to me is the profile of $P(f)$ against $f$ where $P(f_0)$ is the probability density of $f_0$.
Whilst I (think I) can write down that
$$P(f_0) = \frac{1}{V} \int \delta(f_0-f) \rm{d}xyz,$$
with $V$ as the volume of my region,
This in an incredibly difficult integral that I really have no hope of computing.
The specific function I'm dealing with is
$$f(x,y,z) = \frac{1}{(x^2 + y^2+z^2)^{3/2}}\left(1-\frac{3x^2}{x^2 + y^2+z^2}\right)$$.
The region is a circle of sphere r but centred at $(x_0,0,0)$ and not at the origin.