设 $f(x, y)$ 为连续函数, 则 $\int_{0}^{\frac{\pi}{4}} \mathrm{~d} \theta \int_{0}^{1} f(r \cos \theta, r \sin \theta) r \mathrm{~d} r$ 等于 $(\quad)$
A. $\int_{0}^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_{x}^{\sqrt{1-x^{2}}} f(x, y) \mathrm{d} y$.
B. $\int_{0}^{\frac{\sqrt{2}}{2}} \mathrm{~d} x \int_{0}^{\sqrt{1-x^{2}}} f(x, y) \mathrm{d} y$.
C. $\int_{0}^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_{y}^{\sqrt{1-y^{2}}} f(x, y) \mathrm{d} x$.
D. $\int_{0}^{\frac{\sqrt{2}}{2}} \mathrm{~d} y \int_{0}^{\sqrt{1-y^{2}}} f(x, y) \mathrm{d} x$.