 Solution

The parallel axis theorem states $I=I_G+M{d}^2$ for parallel axes separated by $d$, with the reference axis through the centre of mass. The perpendicular axis theorem states $I_z=I_x+I_y$ for a planar lamina in the $xy$ plane, about mutually perpendicular axes through one point. It does not apply directly to a three-dimensional solid.

For the transverse moment of inertia of a uniform solid cylinder, divide the cylinder into thin disks at axial positions $z\in [-h/2,h/2]$. Symmetry and the perpendicular axis theorem give the diametral moment of inertia $r^{2}dM/4$ of each disk; the parallel axis theorem adds $z^{2}dM$. HenceThe thin-disk limit is $M{r}^2/4$, while the slender-rod limit is $M{h}^2/12$.