How to calculate the moment of inertia of a cut out portion? So, I have watched a lot of videos on YouTube, whereby if there is a portion cut out from a particular shape (mind you the that the shape is more often that not symmetric), then the moment of inertia is just calculated by subtracting the moment of inertia of the smaller piece from the initial moment of inertia (of the entire shape). I want to know why this can be done. Can it be mathematically proved?
Secondly, in some other lectures that I watched the person would say 
moment of cut out piece about the center of mass of the initial shape, is equal to moment of remaining piece about the center of mass of the initial shape. I want to know why this is the case. 
 A: The first part is really simple. It just follows from properties of integrals. If volume region $R$ is made up of two regions $R_A$ and $R_B$ then the moment of inertia about some axis is given by$^*$
$$I=\int_R r^2\,\text dm==\int_{R_1} r^2\,\text dm+\int_{R_2} r^2\,\text dm$$
The second part doesn't make sense to me. If you cut a really small piece out of object then certainly the piece's moment of inertia is not equal to that of the remaining part. Either you aren't explaining it well, you misread that point, or your source is just incorrect. 
As a very simple counterexample, consider an object that has a non-zero moment of inertia about its center of mass (which is true of any object, but I want to explicitly make this point for this example). Then just cut a really small piece from the center of mass of an object (assuming the center of mass is located within three object). The piece's moment of inertia will essentially be $0$, and the moment of inertia for the remaining piece will essentially be equal to that of the original piece. But the original moment of inertia was non-zero. Therefore we have shown your conjecture certainly is not always true.

$^*$ Note this is the same thing as you see in introductory calculus
$$\int_a^cf(x)\,\text dx=\int_a^bf(x)\,\text dx+\int_b^cf(x)\,\text dx$$
