Using the formula for the moment of inertia of a uniform sphere, find the moment of inertia of a thin spherical layer of mass m and radius R relative to the axis passing through its centre.

Explanation

Moment of inertia of the shaded portion, about the axis passing through it's centre,
`I=2/5(4/3piR^3rho)R^2-2/5(4/3pir^3rho)r^3`
`=2/5*4/3pirho(R^5-r^5)`
Now, if `R=r+dr`, the shaded portion becomes a shell, which is the required shape to calculate the moment of inertia.
Now, `I=2/5-4/3pirho{(r+dr)^5-r^5}`
`=2/5*4/3pirho(r^5+5r^4dr+........-r^5)`
Neglecting higher terms.
`=2/3(4pir^2drrho)r^2=2/3mr^2`