I know how to take out the moment of inertia of a sphere about an axis passing through a diameter. My method is the same old one of choosing an elemental sheet and integrating. Can someone please explain to me the things I saw on Mathworld please. The method looks fascinating to me and I wish to learn it and use it for other 3 bodies.

Equations (15) and (16) are using your elemental sheet method.

Equation (17) is using the same method, but with an elemental sphere instead of a flat sheet.

Equation (18) is just a way of writing the moment of inertia for all possible diameters instead of just one … but the method of getting them is the same.

(moment of inertia of a rigid body is a 3x3 tensor, so if you write it in coordinates, it looks like a 3x3 matrix)

If you use three integrals in the exam when you could simplify it to two or one, you will lose marks.

The whole point of dividing a volume into slices (what you call "elemental sheets") is to make the integration easier … you choose a slice that you already know the area of, and that saves you two integrations, and leaves you with only one.

Sorry, but that's hopeless … you have to slice the whole volume in a genuine way.

Oh, thats not the way here. We are given ample time in an exam- specially the Sciences and Maths. And my teacher shall appreciate it if I do the question in numerous ways.

By the way it doesnt matter. Could you please explain it to me? Please! I shall be very grateful.

Oh! That was just a thought! I already had the inkling that it might be wrong!