单选题 由曲面$z = \sqrt{x^{2} + y^{2}}$与曲面$z = 2 - x^{2} - y^{2}$所围立体体积等于( )

A、 $\frac{5}{6}\pi$
B、 $\pi$
C、 $\frac{7}{3}\pi$
D、 $2\pi$
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单选题 设$$D$$由$$x^{2} + y^{2} \leq 7$$围成,则$$\iint_{D} e^{-x^{2} - y^{2}} dxdy =$$( )

A、$$\frac{7}{2}(e^{-7} - 1)$$;
B、$$\pi(1 - e^{-7})$$;
C、$$\pi(1 - e^{-49})$$;
D、$$\pi(e^{-7} - 1)$$;

单选题 设$$D$$由抛物线$$y = x^{2}$$直线$$y = 1$$所围成的闭区域在第一象限的部分,则$$\iint_{D} x^{5} y d\sigma$$的值为:

A、$$\frac{1}{30}$$
B、$$\frac{2}{15}$$
C、$$\frac{1}{15}$$
D、$$\frac{1}{10}$$

单选题 已知$$\Omega$$是由球面$$x^{2} + y^{2} + z^{2} = 4$$与抛物面$$x^{2} + y^{2} = 3z$$所围成,则$$\iiint_{\Omega} zdV =$$( )。

A、$$\frac{13\pi}{4}$$
B、$$\frac{\pi}{2}$$
C、$$\frac{3\pi}{4}$$
D、$$\frac{\pi}{4}$$

单选题 设$$I = \iint_{D} x^{2} + y^{2} d\sigma$$,其中$$D: 4 \leq x^{2} + y^{2} \leq 9$$,则有:

A、$$I = 20\pi$$
B、$$I < 20\pi$$
C、$$I > 20\pi$$
D、不确定

单选题 设$$D$$为直线$$y = x, y = -1$$及$$x = 1$$所围成的平面区域,则$$\iint_{D} f(x, y) d\sigma =$$。

A、$$\int_{-1}^{1} dx \int_{-1}^{1} f(x, y) dy$$
B、$$\int_{-1}^{1} dy \int_{-1}^{1} f(x, y) dx$$
C、$$\int_{0}^{1} dy \int_{0}^{1} f(x, y) dx$$
D、$$\int_{-1}^{1} dx \int_{-1}^{1} f(x, y) dy$$

单选题 闭区域$$D = \{(x, y) | x^{2} + y^{2} \leq 2y\}$$的极坐标表示为:( )

A、$$D = \{(\rho, \theta) | 0 \leq \rho \leq 2\theta, 0 \leq \theta \leq \frac{\pi}{2}\}$$
B、$$D = \{(\rho, \theta) | 0 \leq \rho \leq 1, 0 \leq \theta \leq \pi\}$$
C、$$D = \{(\rho, \theta) | 0 \leq \rho \leq 2\sin\theta, 0 \leq \theta \leq \pi\}$$
D、$$D = \{(\rho, \theta) | 0 \leq \rho \leq \sin\theta, 0 \leq \theta \leq \pi\}$$

单选题 将$$I = \iint_{D} e^{-x^{2} - y^{2}} d\sigma$$(其中$$D = \{(x, y) | x^{2} + y^{2} \leq 1\}$$)化为极坐标系下的二次积分,其形式为( )。

A、$$I = 2\int_{0}^{\pi/2} d\theta \int_{0}^{1} e^{-\rho^{2}} \rho d\rho$$
B、$$I = 4\int_{0}^{\pi/2} d\theta \int_{0}^{1} e^{-\rho^{2}} d\rho$$
C、$$I = \int_{0}^{2\pi} d\theta \int_{0}^{1} e^{-\rho^{2}} \rho d\rho$$
D、$$I = \int_{0}^{2\pi} d\theta \int_{0}^{1} e^{-\rho^{2}} d\rho$$

单选题 设$$D = \{(x, y) | x^{2} + y^{2} \leq R^{2}\}$$,则$$\iint_{D} (x^{2} + y^{2}) d\sigma =$$( )

A、$$\frac{1}{3}\pi R^{4}$$
B、$$\pi R^{4}$$
C、$$\frac{2}{3}\pi R^{4}$$
D、$$\frac{1}{2}\pi R^{4}$$