问题
如果积分区域 $D$ 关于 $x$ 轴对称，且积分区域 $D_{1}$ 为积分区域 $D$ 上在 $y$ $\geq$ $0$ 的部分，则以下对二重积分 $\iint_{D}$ $f(x, y)$ $\mathrm{d} \sigma$ 的化简，正确的是哪个？选项
[A]. $\iint_{D} f(x, y) \mathrm{d} \sigma$ $=$ $\left\{\begin{array}{ll} 0, & f(x,-y)=-f(x, y), \\ 2 \iint_{\frac{D}{2}} f(x, y) \mathrm{d} \sigma, & f(x,-y)=f(x, y) \end{array}\right.$[B]. $\iint_{D} f(x, y) \mathrm{d} \sigma$ $=$ $\left\{\begin{array}{ll} 0, & f(x,-y)=-f(x, y), \\ 2 \iint_{D_{1}} f(x, y) \mathrm{d} \sigma, & f(x,-y)=f(x, y) \end{array}\right.$
[C]. $\iint_{D} f(x, y) \mathrm{d} \sigma$ $=$ $\left\{\begin{array}{ll} 0, & f(x,-y)=-f(x, y), \\ \frac{1}{2} \iint_{D_{1}} f(x, y) \mathrm{d} \sigma, & f(x,-y)=f(x, y) \end{array}\right.$
[D]. $\iint_{D} f(x, y) \mathrm{d} \sigma$ $=$ $\left\{\begin{array}{ll} 1, & f(x,-y)=-f(x, y), \\ 2 \iint_{D_{1}} f(x, y) \mathrm{d} \sigma, & f(x,-y)=f(x, y) \end{array}\right.$
$\iint_{D} f(x, y) \mathrm{d} \sigma$ $=$ $\left\{\begin{array}{ll} 0, & f(x,-y)=-f(x, y), \\ 2 \iint_{D_{1}} f(x, y) \mathrm{d} \sigma, & f(x,-y)=f(x, y) \end{array}\right.$