The total kinetic energy (as viewed from one inertial frame) of a free, rigid body is the sum of all the infinitesimal kinetic energies of the components that comprise the body. How do we prove that for a rotating body [tex]E_k=\frac{1}{2}\left(M_{T} v_{c}^{2} + I_{c} ω^{2}\right)[/tex]
Integrate ##\int \frac{1}{2}v^2 \rho dV## (in other words, kinetic energy = 1/2m^2 for all infinitesimal m) and split v into components from translation and rotation and you will get the correct result.