 Solution

Assume a positive common density scale ${\rho }_0$ and use the unit-radius coordinates of the source. The cylinder's density depends on distance from its axis. Its mass isThe radial factor in the cylindrical volume element cancels the apparent density singularity.

For the sphere, the spherical volume element givesThe condition is necessary for finite mass; the integral diverges for $n\le -3$. Equating the two finite masses yields $1/(n+3)=2$, and henceThis satisfies $n>-3$. A divergent pointwise density can therefore have finite total mass: the integrable power-law density at an axis or center must be tested together with its volume factor. Dimensionally, the density laws can be understood as ${\rho }_0(r/R{)}^{-1}$ and ${\rho }_0(r/R{)}^n$ with unit radius $R=1$ in the chosen length units.