问题
已知，平面薄片 $D$ 的面密度为 $\rho(x, y)$, 若 $\rho(x, y)$ 在 $D$ 上连续，则，薄片的质心坐标 $(\bar{x}, \bar{y})$ 为多少？选项
[A]. $\bar{x}$ $=$ $\frac{\iint_{D} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} x \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} y \rho(x, y) \mathrm{d} \sigma}$[B]. $\bar{x}$ $=$ $\frac{\iint_{D} x \rho^{\prime}(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y \rho^{\prime}(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$
[C]. $\bar{x}$ $=$ $\frac{\iint_{D} x^{\prime} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y^{\prime} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$
[D]. $\bar{x}$ $=$ $\frac{\iint_{D} x \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$
$\bar{x}$ $=$ $\frac{\iint_{D} x \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$