设 $f(x, y)$ 是连续函数，则 $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \mathrm{~d} x \int_{\sin x}^1 f(x, y) \mathrm{d} y=(\quad)$
$\text{A.}$ $\int_{\frac{1}{2}}^1 \mathrm{~d} y \int_{\frac{\pi}{6}}^{\arcsin y} f(x, y) \mathrm{d} x$
$\text{B.}$ $\int_{\frac{1}{2}}^1 \mathrm{~d} y \int_{\arcsin y}^{\frac{\pi}{2}} f(x, y) \mathrm{d} x$
$\text{C.}$ $\int_0^{\frac{1}{2}} \mathrm{~d} y \int_{\frac{\pi}{6}}^{\arcsin y} f(x, y) \mathrm{d} x$
$\text{D.}$ $\int_0^{\frac{1}{2}} \mathrm{~d} y \int_{\arcsin y}^{\frac{\pi}{2}} f(x, y) \mathrm{d} x$