Center of Mass

By dividing a solid body into small parts with the mass $\Delta m_i$ of each part $i$ and the position vector $\vec{r}_i$ pointing to that part, the center of mass can be calculated according to $$\vec{r}_\mathrm{CM} = \frac{\sum\Delta m_i\vec{r}_i}{\sum \Delta m_i}$$ Shrinking these parts to infinitesimal small sizes leads to the replacement of the sums by integrals over the whole volume $V$: $$\vec{r}_\mathrm{CM} = \frac{1}{M}\int\vec{r}\,\mathrm{d}m$$ If one replaces $m$ with the product $V\varrho$, where the density $\varrho$ can also depend on the position $\vec{r}$, this integral can be written as $$\boxed{\vec{r}_\mathrm{CM} = \frac{1}{M}\int\vec{r}\varrho(\vec{r})\,\mathrm{d}V}$$ For better understanding the structure of this integral, it can be spilitted into three equations for each component: $$x_\mathrm{CM} = \frac{1}{M}\int x\varrho(x,y,z)\,\mathrm{d}V$$ $$y_\mathrm{CM} = \frac{1}{M}\int y\varrho(x,y,z)\,\mathrm{d}V$$ $$z_\mathrm{CM} = \frac{1}{M}\int z\varrho(x,y,z)\,\mathrm{d}V$$