My calculus book defines the moments of inertia of a solid with respect to the coordinate axes, as:
and the moment of inertia with respect to the origin as
The $I_0$ definition looks inconsistent with the previous ones Trying to prove it, I get: $I_0=I_x+I_y+I_z = \int_\Omega 2(x^2+y^2+z^2) \mu(x,y,z)dxdydz$ Besides I can't figure out what the physical interpretation of moment of inertia of a solid with respect to a point is. How do I make sense of it?
You can find your answer in how Moment of Inertia is defines for a point and an axis. Nonetheless, it's the integral of the product of distance of infinitesimally small points on/inside a solid and the mass of those points. To be precise,
Moment of Inertia= $\int_{\Omega}$(distance of the section from the point/line)*(mass of the infinitesimally small section)
Now the mass density is constant for any solid. But the distance :
From origin = $x^2+y^2+z^2$
From X-Axis = $y^2+z^2$
From Y-Axis = $x^2+z^2$
From Z-Axis = $x^2+y^2$
You can clearly see how $I_0 = 1/2(I_x+I_y+I_z)$ unlike the book.