Why spherical shells give wrong moment of inertia for a solid sphere

[david456103]

For calculating I of a uniform solid sphere, why can't we use thin spherical shells? When I try to use spherical shells I get (3/5)MR^2. Every single derivation uses thin cylindrical shells and end up with the correct expression((2/5)MR^2) but they never explain why it is correct to use cylindrical shells and not spherical shells.
Thanks!

 

[haruspex]

You certainly can get there via spherical shells, but it produces the same answer: 2/5.
What do you get for the MI of the shell?