> [!NOTE] Definition (Centre of mass of solid) > Consider a solid occupying a region $\Omega \subset \mathbb{R}^{3}$ with density $\rho(x,y,z)$ and total mass $M.$ The coordinates of its **centroid**, $(\bar{x},\bar{y},\bar{z})$ are given by the [[Integration in Cartesian coordinates|triple integrals]]: $\bar{x}=\frac{1}{M} \int \int \int_{\Omega} \rho x \, dV, \quad \bar{y} = \frac{1}{M} \int \int \int_{\Omega} \rho y \, dV, \quad \bar{z}= \frac{1}{M} \int \int \int_{\Omega} \rho z \, dV$ # Examples - [[Integration in Cartesian coordinates#^116ced|Centroid of tetrahedron]]. - [[Integration in cylindrical coordinates]]