Center of Mass

we know the torque of a body is $\tau =F\times r$, f is force and  r is vector to the origin.
imagine a horizontal strip with 3 point on it.$\left\{ x_ 1,x_ 2,x_3 \right\}$ with masses :$\left\{ m_ 1,m_ 2,m_3 \right\}$ and a fulcrum $\overset { \_  }{ x }$ on the x-axis origin.
the system might balance, or might not, it depends on how masses are arranged.

each point $m_k$ exert a force toward the ground,according to the gravity force. the force is $m_{ k }.g$;
and the d vector in here is $x_k$. so torque is $\tau =m_{ k }.g\quad \times \quad x_{ k }$;
we know how to balance the system, we should place the fulcrum where the sum of torques would be zero.

$$F=m.g\\ r=x-\overset { \_  }{ x } \\ \\ \sum { \tau _{ k } } =\sum { F_{ k }\times r_{ k } } =0\\ \sum { m_{ k }.g.(x_{ k }-\overset { \_  }{ x } ) } =0\\ g\sum { m_{ k }.(x_{ k }-\overset { \_  }{ x } ) } =0\\ \sum { m_{ k }.x_{ k }-m_{ k }\overset { \_  }{ x }  } =0\\ \sum { m_{ k }.x_{ k } } -\sum { m_{ k }\overset { \_  }{ x }  } =0\\ \sum { m_{ k }.x_{ k } } =\sum { m_{ k }\overset { \_  }{ x }  } \\ \overset { \_  }{ x } =\frac { \sum { m_{ k }.x_{ k } }  }{ \sum { m_{ k } }  } \\ $$

we call the numerator "system moment" and denominator "system mass'.
imagine a strip from x=a to x=b, we can divide the stripe to small piece of mass $\Delta m$
then we can say the $\overset { \_  }{ x }$ is approximately equal :

$$\overset { \_  }{ x } \approx \frac { \sum { \Delta m_{ k }.x_{ k } }  }{ \sum { m_{ k } }  }$$
if the density of strip at $x_k$  is $\sigma (x_k)$ then we can write :
$$\Delta m_{ k }=\sigma (x_{ k }).\Delta x_{ k }\\ \\ \overset { \_  }{ x } \approx \frac { \sum { x_{ k }.\sigma (x_{ k })\Delta x_{ k } }  }{ \sum { \sigma (x_{ k })\Delta x_{ k } }  } \\ $$

the numerator and denominator are a Riemann sum of density function. we can write it as integral :
$$\overset { \_  }{ x } =\frac { \int _{ a }^{ b }{ x\sigma (x)dx }  }{ \int _{ a }^{ b }{ \sigma (x)dx }  }$$