How to devise moment of inertia formula of solid sphere?

[imadrea]

Homework Statement

how to divide moment of inertia of solid sphere about its central axis?. Solid sphere has radius R, mass M.

Homework Equations

I=∫r²dm
2/5 MR^2

The Attempt at a Solution

https://photos.google.com/search/_tra_/photo/AF1QipPoXyad0q1Y3yisc0LeeJHGApkIrGbitK6kAk5p
i try to imagine that solid sphere is a group of infinite disk. a disk have volume dv=πr²dx.
dm=ρdv=πρr²dx.
I=^R∫_-Rr²πρr²dx
I=πρ^R∫_-Rr⁴dx
I=πρ^R∫_-R(R²-x²)²dx
I=πρ^R∫_-RR⁴-2R²x²+x⁴dx
I=πρ[2R⁵-⁴/³R⁵+2/5R⁵]
!=πρR⁵[30-20+6]/15
I=16/15 πρR⁵
I=4/5 (4/3πρR³)R²
I=4/5MR²

where my eror? i have spent 2 days to solve it but i am failed until now.

 

[haruspex]

In your first integral, you introduce an extra factor r². I assume this is related to the moment of inertia of a disk about its axis. Have you forgotten something there?

 

[imadrea]

I assume this is related to the moment of inertia of a disk about its axis. Have you forgotten something there?

i think this is just subtitution for dm=ρπr²dx. I'm confused

 

[haruspex]

i think this is just subtitution for dm=ρπr²dx. I'm confused

No, in the next line you have another r² factor.