=2/3μ∫∫∫(x^2+y^2+z^2)dxdydz

=2μ/3∫[0,2π]dθ∫[0,π]dφ∫[0,r]ρ^2*ρ^2sinφdρ

=8πμr^5/ 15

=2/5r^2(4π/3μr^3)

=2/5mr^2

Note: ∫[a, b]f(x)dx means the definite integral of f(x) on [a, b]. μ represents the density here.

I don't know what the integral formula you wrote means. Can you make it clear?

You still haven't made it clear. Judging from your formula, I still don't understand what your thinking is like after thinking for a long time. I haven't figured out what you can do with one-dimensional integration for a long time, even if you use slice method, you have to integrate twice. What kind of division method is used (how to divide the ball or which part dm refers to) and which axis is used as the rotation axis? Only when I make it clear can I know what your problem is.

If it is divided into disks, there is basically no place to write your expression correctly. Use at least one double integral: first find the moment of inertia of each disk, and then add up the moments of inertia of all disks.

For each optical disc

dJ=μ∫[0,√(r^2-x^2)]2πhdhdx*h^2

=μdx∫[0,√(r^2-x^2)]2πh^3dh

=πμ(r^2-x^2)^2/2dx

Where 2π 2π 2πhdhdx represents the volume of a small ring which is h from the X axis on the disk.

therefore

j=∫[-r,r]dj=πμ∫[-r,r](r^2-x^2)^2/2dx

=8πμr^5/ 15

=2mr^2/5