A hollow cylinder  of density $\rho$ of has length L inner radius  a and outer radius b.  its moment of inertia about the axis of the cylinder is

Given length= L
Inner radius= a
Outer radius=  b
 M.I. of a shell of thickness dr  and radius r  is 
$\begin{array}{l}\text{d}I=\left(\rho 2\text{π}rLdr\right)r^2\text{}\\ I=\underset{a}{\overset{b}{ }}\text{d}I=\underset{a}{\overset{b}{ }}\rho 2\text{π}rLdr{r}^2=2\text{π}\rho L\underset{a}{\overset{b}{ }}{r}^3\text{d}r\text{}=\frac{2\text{π}\rho L\left(b^4-a^4\right)}{4}\end{array}$