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

Given length= LInner radius= aOuter radius=  b M.I. of a shell of thickness dr  and radius r  is dI=ρ2πrLdrr2​I= abdI= abρ2πrLdr r2=2πρL abr3dr​=2πρLb4−a44