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 $\textcolor{orange}{\bar{x}}$ 和纵坐标 $\textcolor{orange}{\bar{y}}$ 分别是多少？选项
[A]. $\begin{cases} & \bar{x} = \frac{\int_{L} x \rho \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} y \rho \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \end{cases}$[B]. $\begin{cases} & \bar{x} = \frac{\int_{L} x \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} y \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \end{cases}$
[C]. $\begin{cases} & \bar{x} = \frac{\int_{L} x^{2} \rho \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} y^{2} \rho \mathrm{d} s}{\int_{L} \rho \mathrm{d} s} \end{cases}$
[D]. $\begin{cases} & \bar{x} = \frac{\int_{L} \rho \mathrm{d} s}{\int_{L} x \rho \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} \rho \mathrm{d} s}{\int_{L} y \rho \mathrm{d} s} \end{cases}$
$\begin{cases} & \textcolor{orange}{\bar{x}} = \frac{\int_{L} \textcolor{red}{x} \textcolor{green}{\cdot} \textcolor{red}{\rho} \mathrm{d} s}{\int_{L} \textcolor{red}{\rho} \mathrm{d} s} \\ & \textcolor{orange}{\bar{y}} = \frac{\int_{L} \textcolor{red}{y} \textcolor{green}{\cdot} \textcolor{red}{\rho} \mathrm{d} s}{\int_{L} \textcolor{red}{\rho} \mathrm{d} s} \end{cases}$