Using centroid of object to find moment of inertia

[petitericeball]

Hey, I was doing some problems involving finding the moment of inertia of a spun spandrel, and I came across the idea of using the centroid to find the moment.

For example, if you have a parabola, find the centroid. If you're rotating around the x-axis (y=x^2), then find y_bar and multiply the total mass by y_bar^2 (or x_bar^2 for Iyy). This isn't working, but it seems like it should. Can anybody explain why?

 

[SteamKing]

Maybe because the centroid is based on the first moment of area or volume (x*A or x*V) and the moment of inertia is based on the second moment of area or volume (x^2*A or x^2*V). For example, objects which have an axis of symmetry will have a zero first moment about that axis, so that the centroid will lie on the axis (for bodies with constant rho at least). On the other hand, second moments about the same axis of symmetry will be non-zero (due to the presence of the x^2 term).