rotational dynamics - Moment of Inertia for Arbitrary Shape

Consider any arbitrary shape with arbitrary rotation axes, global and or local to the shape that are able to describe any orientation of the shape. The shape is also defined with an arbitrary density function. Using

$$dI=r^2dm$$

to obtain the moment of inertia in each of those axes, would then

$$\sum \frac{1}{2}I_i\omega_i^2$$

equal the total rotational kinetic energy of the shape?