Can you explain the notation $\frac{\int dm \vec{r}}{\int dm}$?

This is a generalization of the discrete formula for center of mass, $\vec{R}_{COM} = \frac{1}{M} \sum_i m_i \vec{r}_i$, where $M=\sum_i m_i$. Think of the continuous mass distribution of being made up of an infinite number of infinitesimally small pieces (one of the basic ideas of calculus), each of mass $dm$. When you turn a discrete sum into an integral, the $m_i$'s become $dm$ and the sum turns into an integral. The denominator is $\int dm$ which is just the total mass $M$; it's the sum of all the infinitesimal pieces $dm$.