设 $f(x)$ 是连续的奇函数， $g(x)$ 是连续的偶函数，区域
$$
D=\{(x, y) \mid 0 \leq x \leq 1,-\sqrt{x} \leq y \leq \sqrt{x}\}
$$
则下列结论正确的是
$\text{A.}$ $\iint_D f(y) g(x) \mathrm{d} x \mathrm{~d} y=0$
$\text{B.}$ $\iint_D f(x) g(y) \mathrm{d} x \mathrm{~d} y=0$
$\text{C.}$ $\iint_D[f(x)+g(y)] \mathrm{d} x \mathrm{~d} y=0$
$\text{D.}$ $\iint_D[f(y)+g(x)] \mathrm{d} x \mathrm{~d} y=0$