Moment of Non-Uniform Disk

Differential area:

$dA = r \, dr \, d \theta$

Differential mass:

$dm = \rho \, dA = (r \, \theta) \, r \, dr \, d \theta\, = r^2 \, \theta \, dr \, d \theta$

Differential moment:

$dI = dm \, r^2 = r^4 \, \theta \, dr \, d \theta$

Total moment:

$I = \int_0^{2 \pi} \int_0^1 \, r^4 \, \theta \, dr \, d \theta \\ = \int_0^1 \, r^4 \, dr \int_0^{2 \pi} \theta \, d \theta \\ = \frac{1}{5} \frac{4 \pi^2}{2} \\ = \frac{2}{5} \pi^2$