I have a density function $\rho(r): \mathbb{R}^3 \mapsto \mathbb{R}^+$ depending on the radial coordinate $r$ in spherical coordinates. I need to integrate it with respect to the variable $z$, for example $\int_{-\infty}^{+\infty} \rho(r)\, dz$. However, the metric in $\mathbb{R}^{3}$ is non-Euclidean and in spherical coordinates is given by: $$ ds^2 = A(r)\, dr^2 + r^2(d\theta^2 + \sin^2\theta\, d\phi^2), $$
where $A(r)$ is a smooth function depending only on the radial coordinate.
How to properly define cylindrical coordinates $(R,\phi,z)$ in order to perform the $z$-integration?
Intuitively I would guess: $$A(r)r^2+A(R)R^2=z^2$$ but I’m not sure. Any suggestions?
Rephrasing: how to compute the surface mass of a disk perpendicular to the z-axis? With the Euclidean metric, one has the famous Abel de-projection formula. How this last is modified due to the presence of the factor $A(r)$?