Explain the theoretical method for the estimation of the centre of mass of a solid body.

(N/A) solid body is composed of microscopic particles (molecules,ions,atoms) distributed continuously throughout its volume.
As shown in the figure,consider a solid body divided into small mass elements,each having a mass $dm$ and a position vector $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$.
For a system of discrete particles with masses $m_1, m_2, \ldots, m_n$ at position vectors $\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_n$,the position vector of the centre of mass $\vec{R}$ is given by:
$\vec{R} = \frac{\sum_{i=1}^{n} m_i \vec{r}_i}{\sum_{i=1}^{n} m_i}$
For a continuous solid body,we replace the summation with an integral. Let $M$ be the total mass of the body,where $M = \int dm$.
The position vector of the centre of mass is:
$\vec{R} = \frac{1}{M} \int \vec{r} dm$
In terms of Cartesian coordinates $(X, Y, Z)$,this can be expressed as:
$X = \frac{1}{M} \int x dm$
$Y = \frac{1}{M} \int y dm$
$Z = \frac{1}{M} \int z dm$
These equations allow us to calculate the centre of mass for any solid body by integrating over its entire volume.