问题
若平面图形 $D$ 的线密度为 $\rho(x, y)$, 则该平面图形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]. $\begin{cases} & \bar{x} = \frac{\iint_{D} x^{2} \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \\ & \bar{y} = \frac{\iint_{D} y^{2} \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \end{cases}$[B]. $\begin{cases} & \bar{x} = \frac{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} x \rho(x, y) \mathrm{d} x \mathrm{d} y} \\ & \bar{y} = \frac{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} y \rho(x, y) \mathrm{d} x \mathrm{d} y} \end{cases}$
[C]. $\begin{cases} & \bar{x} = \frac{\iint_{D} x \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \\ & \bar{y} = \frac{\iint_{D} y \rho(x, y) \mathrm{d} x \mathrm{d} y}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \end{cases}$
[D]. $\begin{cases} & \bar{x} = \frac{\iint_{D} x \mathrm{d} x}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \\ & \bar{y} = \frac{\iint_{D} y \mathrm{d} y}{\iint_{D} \rho(x, y) \mathrm{d} x \mathrm{d} y} \end{cases}$
$\begin{cases} & \textcolor{orange}{\bar{x}} = \frac{\iint_{D} \textcolor{red}{x} \textcolor{green}{\cdot} \textcolor{red}{\rho}(x, y) \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y}}{\iint_{D} \textcolor{red}{\rho}(x, y) \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y}} \\ & \textcolor{orange}{\bar{y}} = \frac{\iint_{D} \textcolor{red}{y} \textcolor{green}{\cdot} \textcolor{red}{\rho}(x, y) \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y}}{\iint_{D} \textcolor{red}{\rho}(x, y) \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y}} \end{cases}$