## Essential University Physics: Volume 1 (4th Edition)

$\frac{3MR^2}{10}$
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: $I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \int_0^h z^4$ $I=\frac{R^2}{2h^2} \cdot \frac{3M}{h^3} \frac{h^5}{5}$ $I=\frac{3MR^2}{10}$