# 面密度　$\rho \text{\hspace{0.17em}}〔{\text{kg/m}}^{\text{2}}〕$ が一定の剛体の重心

ここで，有限の質量$\Delta {m}_{i}=\rho \Delta {s}_{i}\text{\hspace{0.17em}}〔\text{kg}〕$ ($\Delta {s}_{i}$ は面積)$\left(i=1,2,\cdots ,n\right)$ が　$n$　個集まり，剛体が形成されていると考える．

そこで， $n$ 質点系の重心(2次元)において， $n\to \infty$ の極限をとると，

${x}_{G}=\underset{n\to \infty }{\mathrm{lim}}\frac{\Delta {m}_{1}{x}_{1}+\Delta {m}_{2}{x}_{2}+\cdots +\Delta {m}_{n}{x}_{n}}{\Delta {m}_{1}+\Delta {m}_{2}+\cdots +\Delta {m}_{n}}$
$=\underset{n\to \infty }{\mathrm{lim}}\frac{\rho \Delta {s}_{1}{x}_{1}+\rho \Delta {s}_{2}{x}_{2}+\cdots +\rho \Delta {s}_{n}{x}_{n}}{\rho \Delta {s}_{1}+\rho \Delta {s}_{2}+\cdots +\rho \Delta {s}_{n}}$
$=\underset{n\to \infty }{\mathrm{lim}}\frac{{x}_{1}\Delta {s}_{1}+{x}_{2}\Delta {s}_{2}+\cdots +{x}_{n}\Delta {s}_{n}}{\Delta {s}_{1}+\Delta {s}_{2}+\cdots +\Delta {s}_{n}}$
$=\underset{n\to \infty }{\mathrm{lim}}\frac{\sum _{i=1}^{n}{x}_{i}\Delta {s}_{i}}{\sum _{i=1}^{n}\Delta {s}_{i}}=\underset{n\to \infty }{\mathrm{lim}}\frac{1}{s}\sum _{i=1}^{n}{x}_{i}\Delta {s}_{i}=\frac{1}{s}\underset{n\to \infty }{\mathrm{lim}}\sum _{i=1}^{n}{x}_{i}\Delta {s}_{i}$

$\therefore \text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{G}=\frac{1}{s}\int x\text{\hspace{0.17em}}ds\text{\hspace{0.17em}}〔\text{m}〕$

$\therefore \text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{G}=\frac{1}{s}\int y\text{\hspace{0.17em}}ds\text{\hspace{0.17em}}〔\text{m}〕$