设函数 $f(u)$ 连续，区域 $D=\left\{(x, y) \mid x^2+y^2 \leqslant 2 y\right\}$ ，则 $\iint_D f(x y) \mathrm{d} x \mathrm{~d} y$ 等于 ()
A. $\int_{-1}^1 \mathrm{~d} x \int_{-\sqrt{1-x^x}}^{\sqrt{1-x^x}} f(x y) \mathrm{d} y$ ．
B. $2 \int_0^2 \mathrm{~d} y \int_0^{\sqrt{2 y-y^2}} f(x y) \mathrm{d} x$ ．
C. $\int_0^\pi \mathrm{d} \theta \int_0^{2 \sin \theta} f\left(r^2 \sin \theta \cos \theta\right) \mathrm{d} r$ ．
D. $\int_0^\pi \mathrm{d} \theta \int_0^{2 \sin \theta} f\left(r^2 \sin \theta \cos \theta\right) \mathrm{rd} r$ ．