Question

Explain why when deriving the formula for the moment of inertia for a body mass, one must consider the surface mass density when defining one's differential mass element?

Answer 1

The moment of inertia is

where is the differential mass element

The differential mass element is simply density times differential volume element

Note that in general the density is a function of coordinate, r and in that case, you have to include the density function in the integral to get the result.

where the integral is over the total volume V and the differential volume is a function of coordinates depending on the object for which moment of inertia you are evaluating