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 and m is 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 define ds in a good way to calculate this integral. Is it as simple as letting x run from -R to R? No it cant be!