Thanks for the link, although I don't actually have any problems with that derivation. What I don't understand is the why my own approach [just integrating (y^2)(dm) to obtain I ] and the derivation shown in the link [ starting with dI = (1/2)(y^2)(dm) ] are any different.
When finding I for a solid cylinder for example, you would use thin concentric cylindrical shells as the infinitesimal mass elements, but it is not necessary to consider what the moment of inertia of each shell is. In this case you can start with I = integral of (y^2)(dm) to obtain I = (1/2)MR^2. But with a sphere, all the literature suggests that the moment of inertia of each mass element must be considered, since all the derivations start with dI = (1/2)(y^2)(dm).