Where does a shape balance, and how hard is it to spin? Two integrals of the same lamina, answering different questions.
A uniform disk of mass M and radius R has I = ½MR². Try it and check.
Change the density and watch the balance point slide.
Cut a shape out of card and ask two different questions about it. Where does it balance on a pin? And how hard is it to spin? Both are integrals over the same plate, and both weight each little piece by the density there — but they weight position very differently.
The balance point averages position with weight ρ dA: it is the total of x times mass, divided by total mass. Push the density to one side and the balance point slides that way, exactly as intuition says.
The moment of inertia weights by r² ρ dA instead — distance squared from the axis. That square is the whole story. Mass far from the axis counts enormously more than mass near it, which is why a hoop is so much harder to spin than a solid disk of the same weight, and why a figure skater speeds up by pulling her arms in without changing her mass at all.
The circle drawn around the balance point is the radius of gyration: the single distance at which you could put the entire mass, as one point, and get the same resistance to spinning. It is a root-mean-square of distance, so it always sits further out than the average distance does — again, because of the square.