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I was reading this website's derivation of the moment of inertia of a solid sphere here: http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph3

For the life of me I can't see where the (1/2)y

^{2}term comes from. I would understand y

^{2}because that would be the radius, but the 1/2?

I understand the other steps but I'm very much stuck on this first one.

A related question for anyone who has an opinion: this website starts out these problems by writing dI = abc dm. My physics professor tends to start with dm = dfg dr. Is there a best way to approach these problems? I'm always the most confused at the beginning when I need to define the above relationships. Thanks!