Let's say, I have a density function,
$$\rho(r_{e}) \sim (r_e)^{-s} \qquad \text{where} \quad r_{e} = x^{2} + \frac{y^{2}}{p^{2}} + \frac{z^{2}}{q^{2}}$$ and $p$ is the $y$-to-$x$ ratio, while $q$ is the $z$-to-$x$ axis ratio.
I need to perform a volume integral, and know how to do it for the spherical case, i.e., when $p=q=1$, as then the Jacobian is simply:
$$\int\int\int \rho(r) dx dy dz = \int\int\int\rho(r) r^{2} sin(\theta) dr d\theta d\phi$$
Can anyone help me with the transformation between the ellipsoidal radius $r_{e}$ and the spherical $r$? I am guessing it would be a factor of $p$ and $q$, but can't get it.
Apparently, for the two-$d$ case,
$$r dr = (\text{semi-minor/semi-major}) r_{e} dr_{e}$$