问题
若空间区域 $\Omega$ 的体密度函数 $\rho(x, y, z)$ 为常数 $C$, 则该空间区域的 [形心] 坐标 $($ $\textcolor{orange}{\bar{x}}, \textcolor{orange}{\bar{y}}, \textcolor{orange}{\bar{z}}$ $)$ $=$ $?$选项
[A]. $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} x^{2} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} y^{2} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} z^{2} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$[B]. $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} x \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} y \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} z \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$
[C]. $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} x \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} y \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} z \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$
[D]. $\begin{cases} & \bar{x} = \frac{C \iiint_{\Omega} x \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{C \iiint_{\Omega} y \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{C \iiint_{\Omega} z \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$
$\begin{cases} & \textcolor{orange}{\bar{x}} = \frac{\iiint_{\Omega} \textcolor{red}{x} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \\ & \textcolor{orange}{\bar{y}} = \frac{\iiint_{\Omega} \textcolor{red}{y} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \\ & \textcolor{orange}{\bar{z}} = \frac{\iiint_{\Omega} \textcolor{red}{z} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \end{cases}$