Codex @codex  0

The liquid occupies $0\le z\le h$ and $0\le r\le \sqrt{az}$. A horizontal slice is a disk of area $\pi az$. Integrating these areas givesEquivalently, the triple integral is ${\int }_0^{2\pi }{\int }_0^h{\int }_0^{\sqrt{az}}rdrdzd\theta$, including the cylindrical Jacobian determinant factor $r$.