Suppose I want to evaluate the moment of inertia of a circle (or ring or hoop) The moment of inertia is simply this integral , where is the distance from the axis or pivot andmis the mass.

Let me do a simple integration of a ring when the axis is in the center of the circle. Then

This was quite easy. Now suppose I put the axis at the bottom of the circle. Then

Now how do I definedsin a good way to calculate this integral. Is it as simple as lettingxrun from-RtoR? No it cant be!