Center of Mass Suppose we have a finite collection of masses in the coordinate plane, the mass $m_{k}$ located at the point $\left(x_{k}, y_{k}\right)$ as shown in the figure.
Each mass $m_{k}$ has moment $m_{k} y_{k}$ with respect to the $x$ -axis and moment $m_{k} x_{k}$ about the $y$ -axis. The moments of the entire system with respect to the two axes are
$$\begin{array}{l}{\text { Moment about } x \text { -axis: } M_{x}=\sum m_{k} y_{k}} \\ {\text { Moment about } y \text { -axis: } M_{y}=\sum m_{k} x_{k}}\end{array}$$
The center of mass is $(\overline{x}, \overline{y})$ where
$$\overline{x}=\frac{M_{y}}{M}=\frac{\sum m_{k} x_{k}}{\sum m_{k}} \quad$ and $\quad \overline{y}=\frac{M_{x}}{M}=\frac{\sum m_{k} y_{k}}{\sum m_{k}}$$
Suppose we have a thin, flat plate occupying a region in the plane.
(a) Imagine the region cut into thin strips parallel to the $y$ -axis. Show that
$$\overline{x}=\frac{\int x d m}{\int d m}$$
where $d m=\delta d A, \delta=$ density (mass per unit area), and $A=$ area of the region.
(b) Imagine the region cut into thin strips parallel to the $x$ -axis. Show that
$$\overline{y}=\frac{\int y d m}{\int d m}$$
where $d m=\delta d A, \delta=$ density, and $A=$ area of the region.