设 $D$ 是第一象限中的曲线 $2xy = 1, 4xy = 1$ 与直线 $y = x, y = \sqrt{3}x$ 围成的平面区域, 函数 $f(x, y)$ 在 $D$ 上连续, 则 $\iint_{D} f(x, y) \mathrm{d}x \mathrm{d}y = (\quad)$ (A) $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\mathrm{d}\theta \int_{\frac{1}{2\sin 2\theta}}^{\frac{1}{\sin 2\theta}}f(r\cos \theta ,r\sin \theta)r\mathrm{d}r.$ (B) $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \mathrm{d}\theta \int_{\frac{1}{\sqrt{2\sin 2\theta}}}^{\frac{1}{\sqrt{\sin 2\theta}}} f(r \cos \theta, r \sin \theta) r \mathrm{d}r.$ ( C) $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\mathrm{d}\theta \int_{\frac{1}{2\sin 2\theta}}^{\frac{1}{\sin 2\theta}}f(r\cos \theta ,r\sin \theta)\mathrm{d}r.$ $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \mathrm{d}\theta \int_{\frac{1}{\sqrt{2\sin 2\theta}}}^{\frac{1}{\sqrt{\sin 2\theta}}} f(r \cos \theta, r \sin \theta) \mathrm{d}r.$