问题
若平面曲线 $L$ 的线密度函数为 $\rho(x)$ 为常数 $C$, 则该平面曲线形心坐标中的横坐标 $\textcolor{orange}{\bar{x}}$ 和纵坐标 $\textcolor{orange}{\bar{y}}$ 分别是多少？选项
[A]. $\begin{cases}& \bar{x} = \frac{\int_{L} x \mathrm{d} s}{\int_{L} \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} y \mathrm{d} s}{\int_{L} \mathrm{d} s} \end{cases}$[B]. $\begin{cases}& \bar{x} = \frac{\int_{L} x^{2} \mathrm{d} s}{\int_{L} \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} y^{2} \mathrm{d} s}{\int_{L} \mathrm{d} s} \end{cases}$
[C]. $\begin{cases}& \bar{x} = \frac{\int_{L} \mathrm{d} s}{\int_{L} x \mathrm{d} s} \\ & \bar{y} = \frac{\int_{L} \mathrm{d} s}{\int_{L} y \mathrm{d} s} \end{cases}$
[D]. $\begin{cases}& \bar{x} = \frac{C \int_{L} x \mathrm{d} s}{\int_{L} \mathrm{d} s} \\ & \bar{y} = \frac{C \int_{L} y \mathrm{d} s}{\int_{L} \mathrm{d} s} \end{cases}$
$\begin{cases}& \textcolor{orange}{\bar{x}} = \frac{\int_{L} \textcolor{red}{x} \textcolor{green}{\cdot} \mathrm{d} s}{\int_{L} \mathrm{d} s} \\ & \textcolor{orange}{\bar{y}} = \frac{\int_{L} \textcolor{red}{y} \textcolor{green}{\cdot} \mathrm{d} s}{\int_{L} \mathrm{d} s} \end{cases}$