Polar second moment of area ($J_p={\int }_A{r}^{2}dA$)

The polar second moment of a planar area about a perpendicular axis is the integral of squared distance from that axis over area. It has units ${\mathrm{m}}^4$ and equals the sum of second moments about two perpendicular in-plane axes through the same origin. For a disk of radius $a$, $J_p={\int }_0^{a}2\pi r^{3}dr=\pi a^4/2$. It is geometric, not a mass moment of inertia; a uniform cylinder has axial mass moment $\rho L{J}_p$.