The trick here is to think of the disk as a collection of a bunch of concentric
rings. Call the mass per unit area of the disk 
. Then we have a bunch
of rings, the inner radius of one of these rings is r and outer radius r+dr.
Then the mass of this ring dm is the surface dA times . So
But dA is just the circumference times dr, 
. So
Now we want to integrate dI over all radii to get I so
Let's write in terms of M and R:
So finally we get
