Match each physical quantity to the triple integral that computes it (over a solid $E$ of density $\rho(x,y,z)$).

• Volume of $E$ → $\iiint_E 1\,dV$
• Mass of $E$ → $\iiint_E \rho\,dV$
• Average density → $\iiint_E \rho\,dV \;/\; \iiint_E 1\,dV$
• Moment about the $xy$-plane → $\iiint_E z\rho\,dV$

One machine, many outputs: change the integrand and $\iiint dV$ measures volume (1), mass ($\rho$), or leverage ($z\rho$, the ingredient of centre-of-mass). Physics quantities are mostly "integrate the right weighting over the body".