平面曲线的质心公式（B007）

问题

若平面曲线 $L$ 的线密度为 $\rho (x)$, 则该平面曲线质心坐标中的横坐标 $\overline{x}$ 和纵坐标 $\overline{y}$ 分别是多少？

选项

[A].   $\{\begin{array}{ll}& \overline{x}=\frac{{\int }_{L}x\mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\\ & \overline{y}=\frac{{\int }_{L}y\mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\end{array}$

[B].   $\{\begin{array}{ll}& \overline{x}=\frac{{\int }_L{x}^2\rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\\ & \overline{y}=\frac{{\int }_L{y}^2\rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\end{array}$

[C].   $\{\begin{array}{ll}& \overline{x}=\frac{{\int }_L\rho \mathrm{d}s}{{\int }_{L}x\rho \mathrm{d}s}\\ & \overline{y}=\frac{{\int }_L\rho \mathrm{d}s}{{\int }_{L}y\rho \mathrm{d}s}\end{array}$

[D].   $\{\begin{array}{ll}& \overline{x}=\frac{{\int }_{L}x\rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\\ & \overline{y}=\frac{{\int }_{L}y\rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\end{array}$

   答 案   

$\{\begin{array}{ll}& \overline{x}=\frac{{\int }_{L}x\cdot \rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\\ & \overline{y}=\frac{{\int }_{L}y\cdot \rho \mathrm{d}s}{{\int }_L\rho \mathrm{d}s}\end{array}$