\(\,\,\,\mathop {{I_0}}\limits_{\left( {MOl\,about\,COM} \right)}  = \int\limits_0^R {{\rho _0}r} \left( {2\pi rdr} \right) \times {r^2} = \frac{{{\rho _0} \times 2\pi {R^5}}}{5}\)

By parallel axis theorem

\(I = {I_0} + M{R^2}\)

\( = \frac{{{\rho _0} \times 2\pi {R^5}}}{5} + \frac{{{\rho _0} \times 2\pi {R^3}}}{3} \times {R^2} = {\rho _0}2\pi {R^5} \times \frac{8}{{15}}\)

\( = M{R^2} \times \frac{8}{5}\)