一. 单选题（共4题，100分）

1. (单选题)设平面区域D:1\leq x^2+y^2\leq 4,f(x,y)是在区域D上的连续函数，则\begin{aligned} \iint_D f(\sqrt{x^2+y^2}){\rm d}x{\rm d}y \end{aligned}等于 (     ).

• A.\begin{aligned} 2\pi\left[\int_0^2rf(r){\rm d}r+\int_0^1rf(r){\rm d}r\right] \end{aligned};

• B.\begin{aligned} 2\pi \int_1^2rf(r^2){\rm d}r \end{aligned} ;

• C.\begin{aligned} 2\pi \left[ \int_0^2rf(r^2){\rm d}r+\int_0^1rf(r^2){\rm d}r \right] \end{aligned};

• D.\begin{aligned} 2\pi\int_1^2rf(r){\rm d}r \end{aligned};

正确答案:D

2. (单选题)把二重积分\begin{aligned} \iint_D{\rm e}^{-x^2-y^2}{\rm d}x{\rm d}y \end{aligned}在极坐标系中化为二次积分，其中D由\begin{aligned} x^2+y^2\leq 1 \end{aligned}所围成(    ).

• A.\begin{aligned} 2\int_0^{\frac {\pi}{2}}{\rm d}\theta\int_0^1r{\rm e}^{-r^2}{\rm d}r \end{aligned};

• B.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^1{\rm e}^{-r^2}{\rm d}r \end{aligned};

• C.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^1r{\rm e}^{-r^2}{\rm d}r \end{aligned}.

• D.\begin{aligned} 4\int_0^{\frac\pi2}{\rm d}\theta\int_0^1{\rm e}^{-r^2}{\rm d}r \end{aligned};

正确答案:C

3. (单选题)设D:\begin{aligned} 1\leq x^2+y^2\leq 4 \end{aligned}，则\begin{aligned} \iint_D\sqrt{x^2+y^2}{\rm d}x{\rm d}y= \end{aligned}(      ).

• A.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^1r^2{\rm d}r \end{aligned};

• B.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_1^4r^2{\rm d}r \end{aligned};

• C.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_1^2r{\rm d}r \end{aligned}.

• D.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_1^2r^2{\rm d}r \end{aligned};

正确答案:D

4. (单选题)设区域\begin{aligned} D= \left\{ (x,y)|x^2+y^2\leq 4 \right\} \end{aligned}，则\begin{aligned} \iint_D\sqrt{x^2+y^2}{\rm d}x{\rm d}y= \end{aligned}(     ).

• A.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^4r^2{\rm d}r \end{aligned};

• B.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_1^2r{\rm d}r \end{aligned}.

• C.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^2r^2{\rm d}r \end{aligned};

• D.\begin{aligned} \int_0^{2\pi}{\rm d}\theta\int_0^2r^3{\rm d}r \end{aligned};

正确答案:C