设 $D$ 是由曲线 $2 x y=1$ 与直线 $x+y=\frac{3}{2}$ 所围成的封闭区域，已知函数 $f(x, y)$ 在区域 $D$ 上连续，则 $\iint_D f(x, y) \mathrm{d} x \mathrm{~d} y=$
A. $2 \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_{\frac{1}{\sqrt{\sin 2 \theta}}}^{\frac{3}{2(\sin \theta+\cos \theta)}} f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$ ．
B. $2 \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \mathrm{~d} \theta \int_{\frac{1}{\sqrt{2 \sin 2 \theta}}}^{\frac{3}{2(\sin \theta+\cos \theta)}} f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$ ．
C. $2 \int_{\frac{\pi}{4}}^{\arctan 2} \mathrm{~d} \theta \int_{\frac{1}{\sqrt{\sin 2 \theta}}}^{\frac{3}{2(\sin \theta+\cos \theta)}} f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$ ．
D. $2 \int_{\frac{\pi}{4}}^{\arctan 2} \mathrm{~d} \theta \int_{\frac{1}{\sqrt{2 \sin 2 \theta}}}^{\frac{3}{2(\sin \theta+\cos \theta)}} f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$ ．