A cylinder with radius R and mass M has density that increases linearly with distance r from the cylinder axis, ρ = αr, where α is a positive constant. Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of M and R.
This is from my textbook, but my textbook has only explained how to find inertias from uniform masses leading up to this question. What's even worse is that this question is given the easiest rating.
I = Ʃ(mi*r^2i)
Normally I would find the integral of r^2*dm, but my book says that it is only for a uniform distribution of mass.
The Attempt at a Solution
I = Ʃ(mi*r^2i) = m(1)*r^2(1) + m(2)*r^2(2) + ...
I just cannot figure out how to incorporate an increasing density into this.