• 2022-06-09
    计算\(\int\!\!\!\int\limits_\sum { { x^2}dydz + {y^2}dzdx + {z^2}} dxdy\),其中\(\sum\)为长方体\(\Omega \)的整个表面外侧,\(\Omega = \{ (x,y,z)|0 \le x \le a,0 \le y \le b,0 \le z \le c\} \)。
    A: \((a + b + c)abc\)
    B: \((a -b + c)abc\)
    C: \((a + b -c)abc\)
    D: \((a - b - c)abc\)
  • A

    内容

    • 0

      设\(D\)是由\( - 1 \le x \le 1 \) ,\( 0 \le y \le 2 \) 所围区域,则\( \int\!\!\!\int\limits_D {\left| {y - {x^2}} \right|} d\sigma \) = \( { { 45} \over {16}} \) 。

    • 1

      曲线$\left\{ \matrix{ {x^2} + {y^2} + {z^2} = 9 \cr y = x \cr} \right.$的参数方程为( ). A: $$\left\{ \matrix{ x = \sqrt 3 \cos t \cr y = \sqrt 3 \cos t \cr z = \sqrt 3 \sin t \cr} \right.(0 \le t \le 2\pi )$$ B: $$\left\{ \matrix{ x = {3 \over {\sqrt 2 }}\cos t\cr y = {3 \over {\sqrt 2 }}\cos t \cr z = 3\sin t \cr} \right.(0 \le t \le 2\pi )$$ C: $$\left\{ \matrix{ x = \cos t\cr y = \cos t\cr z = \sin t \cr} \right.(0 \le t \le 2\pi )$$ D: $$\left\{ \matrix{ x = {{\sqrt 3 } \over 3}\cos t\cr y = {{\sqrt 3 } \over 3}\cos t \cr z = {{\sqrt 3 } \over 3}\sin t\cr} \right.(0 \le t \le 2\pi )$$

    • 2

      设\(D\)为\( 1 \le x \le 2 \) 和\( 0 \le y \le 1 \) 所围区域,则\( \int\!\!\!\int\limits_D { { x^2}{e^{2y}}} d\sigma \) =\( {6 \over 7}\left( { { e^2} - 1} \right) \) 。

    • 3

      函数$y = \arcsin (2x + 1)<br/>$的定义域为 ( ). A: $\{ \left. x \right| - 1 \le x \le 0\} <br/>$ B: $\{ \left. x \right| - \frac{1}{2} \le x \le 0\} <br/>$ C: $\{ \left. x \right|x \ge - \frac{1}{2}\} <br/>$ D: ${\rm{\{ }}\left. x \right|x \le 0\}<br/>$

    • 4

      已知 $X$ 和 $Y$ 的联合密度函数为 $f(x,y)=$ $\begin{cases} cxy,&amp; 0\le x\le 1, 0\le y\le 1,\\ 0,&amp; \text{其他},\end{cases}$,则$c=$______ , $P\{X