Moment of Inertia of a Sphere

[piquante]

So here I just need help understanding a concept.
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)y² term comes from. I would understand y² 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!

 

[therealnihl]

To find the moment of inertia of the sphere, they had to sum the moment of inertia of many tiny disks of mass dm. The moment of inertia of a disk of mass M is (1/2)MR^2. To see a derivation of the moment of inertia of a disk: http://mikebloxham.com/H7A/I for sphere.pdf