Answer
$\frac{3MR^2}{10}$
Work Step by Step
We know:
$I=\int .5r^2 dm$
Thus, we need to find dm:
$dm=\frac{M}{\frac{1}{3}\pi R^2 h}\times\pi r^2 dz$
This can be simplified to:
$dm=\frac{3Mz^2}{h^3}$
We plug this and the equation $r^2 = \frac{R^2z^2}{h^2}$ into the above integral to find:
$I=\int_0^h .5 \frac{R^2z^2}{h^2} \cdot \frac{3Mz^2}{h^3}$
Factoring the constants out of the integral leaves:
$ I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \int_0^h z^4 $
$I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \frac{h^5}{5}$
$I=\frac{3MR^2}{10}$
