Integrable power-law density at an axis or center ($p>-2\text{ at an axis},p>-3\text{ at a point in three dimensions}$)

A nonnegative mass density proportional to $r^p$ has finite local mass near a straight axis if ${\int }_0^{\epsilon }{r}^{p+1}dr$ converges, namely $p>-2$. Near a three-dimensional center the corresponding condition is convergence of ${\int }_0^{\epsilon }{r}^{p+2}dr$, namely $p>-3$. These follow from the cylindrical volume element and spherical volume element. At equality the divergence is logarithmic. An unbounded density does not by itself imply infinite mass.