#2选择题来源：302-2022
$\int_{0}^{2}dy\int_{y}^{2}\frac{y}{\sqrt{1+x^{3}}}\,dx=( )$
- A. $\frac{\sqrt{2}}{6}.$
- B. $\frac{1}{3}.$
- C. $\frac{\sqrt{2}}{3}.$
- D. $\frac{2}{3}.$
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#3选择题来源：303-2024
已知$f(x,y)$连续, 则$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}\mathrm{d}x\int_{\sin x}^{1}f(x,y)\mathrm{d}y=$
- A. $\int_{\frac{1}{2}}^{1}\mathrm{d}y\int_{\frac{\pi}{6}}^{\arcsin y}f(x,y)\mathrm{d}x$
- B. $\int_{\frac{1}{2}}^{1}\mathrm{d}y\int_{\frac{\pi}{2}}^{\arcsin y}f(x,y)\mathrm{d}x$
- C. $\int_{0}^{\frac{1}{2}}\mathrm{d}y\int_{\frac{\pi}{6}}^{\arcsin y}f(x,y)\mathrm{d}x$
- D. $\int_{0}^{\frac{1}{2}}\mathrm{d}y\int_{\frac{\pi}{2}}^{\arcsin y}f(x,y)\mathrm{d}x$
#4选择题来源：301-2025
设函数 $f(x,y)$ 连续, 则 $\int_{-2}^{2} dx \int_{4-x^2}^{4} f(x,y) dy =$
- A. $\int_{0}^{4} \left[ \int_{-2}^{-\sqrt{4-y}} f(x,y)dx + \int_{\sqrt{4-y}}^{2} f(x,y)dx \right] dy$
- B. $\int_{0}^{4} \left[ \int_{-2}^{\sqrt{4-y}} f(x,y)dx + \int_{\sqrt{4-y}}^{2} f(x,y)dx \right] dy$
- C. $\int_{0}^{4} \left[ \int_{-2}^{-\sqrt{4-y}} f(x,y)dx + \int_{2}^{\sqrt{4-y}} f(x,y)dx \right] dy$
- D. $2 \int_{0}^{4} dy \int_{\sqrt{4-y}}^{2} f(x,y)dx$
#4选择题来源：303-2025
设函数$f(x)$连续，则$\int_{0}^{1} dy\int_{0}^{y} f(x)dx=$
- A. $\int_{0}^{1} xf(x)dx$.
- B. $\int_{0}^{1} (1+x)f(x)dx$.
- C. $\int_{0}^{1} (x-1)f(x)dx$.
- D. $\int_{0}^{1} (1-x)f(x)dx$.
#5选择题来源：302-2025
设函数 $f(x,y)$ 连续，则 $\int_{-2}^{2} dx\int_{4-x^{2}}^{4} f(x,y)\,dy=(\ )$
- A. $\int_{0}^{4}\left[\int_{-2}^{-\sqrt{4-y}} f(x,y)\,dx+\int_{\sqrt{4-y}}^{2} f(x,y)\,dx\right]dy$
- B. $\int_{0}^{4}\left[\int_{-2}^{\sqrt{4-y}} f(x,y)\,dx+\int_{\sqrt{4-y}}^{2} f(x,y)\,dx\right]dy$
- C. $\int_{0}^{4}\left[\int_{-2}^{-\sqrt{4-y}} f(x,y)\,dx+\int_{2}^{\sqrt{4-y}} f(x,y)\,dx\right]dy$
- D. $2\int_{0}^{4} dy\int_{\sqrt{4-y}}^{2} f(x,y)\,dx$
#6选择题来源：302-2024
设 $f(x,y)$ 是连续函数，则 $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}dx\int_{\sin x}^{1}f(x,y)\,dy=(\ )$
- A. $\int_{\frac{1}{2}}^{1}dy\int_{\frac{\pi}{6}}^{\arcsin y}f(x,y)\,dx$
- B. $\int_{\frac{1}{2}}^{1}dy\int_{\arcsin y}^{\frac{\pi}{2}}f(x,y)\,dx$
- C. $\int_{0}^{\frac{1}{2}}dy\int_{\frac{\pi}{6}}^{\arcsin y}f(x,y)\,dx$
- D. $\int_{0}^{\frac{1}{2}}dy\int_{\arcsin y}^{\frac{\pi}{2}}f(x,y)\,dx$
#14填空题来源：302-2021
已知函数 $f(t)=\int_{1}^{t^{2}} dx\int_{\sqrt{x}}^{t} \sin\frac{x}{y}\,dy$，则 $f'\left(\frac{\pi}{2}\right)=\underline{\qquad}$.
#17解答题来源：301-2024
已知平面区域 $D = \{(x,y)|\sqrt{1 - y^2} \leq x \leq 1, -1 \leq y \leq 1\}$, 计算 $\iint_{D} \frac{x}{\sqrt{x^2 + y^2}} dxdy$
#17解答题来源：302-2026
计算 $I = \int_{-1}^{1} dx \int_{|x|}^{\sqrt{2-x^2}} y \sin \sqrt{x^2+y^2} dy$
#19解答题来源：302-2022
已知平面区域$D=\{(x,y)\mid y-2\leq x\leq \sqrt{4-y^2},0\leq y\leq 2\}$,计算$I=\iint_D \frac{(x-y)^2}{x^2+y^2}\,dxdy$.
#19解答题来源：303-2025
已知平面有界区域$D = \left\{(x,y) \mid y^{2} \leqslant x, x^{2} \leqslant y\right\}$，计算二重积分$\iint_{D} (x - y + 1)^{2} \mathrm{d}x\mathrm{d}y$.
#20解答题来源：303-2026
设平面区域 $D = \{(x,y) \mid 0 \leq x \leq 1, 0 \leq y \leq 1\}$，计算二重积分
$$\iint_D \dfrac{y}{(1+x^2+y^2)^{\frac{3}{2}}} dx dy.$$