Find the moment of inertia of a cylinder of mass `M`, radius `R` and length `L` about an axis passing through its centre and perpendicular to its symmetry axis. Do this by integrating an elemental disc along the length of the cylinder.

Explanation

Moment of inertia of `n` elemental disc at the distance `x` from axis `1`
`dm=(Mdx)/L`

`MI` of the element disc about axis `1`
`I_(1)=int_(L//2)((dm)/2R^(2)+dmx^(2))`
`=int_(-L//2)^(L//2)M/(2L)R^(3)dx+int_(-L//2)^(L//2)M/Lx^(2)dx`
`=(MR^(2))/2+(ML^(2))/12`