作者impin (pin)
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标题[试题] 99下 王金龙 微积分甲下 第五次小考
时间Wed Jun 8 15:37:45 2011
课程名称︰微积分甲下
课程性质︰必修
课程教师︰王金龙
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2011/05/12
考试时限（分钟）：40
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
A. Let S and T be any disjoint sets whose union has area.
- + -
(a) Show that A (S∪T) ≦ A (S) + A (T) for any finite partition.
n n n
+ - - +
(b) Show that A (S) + A (T) ≦ A (S∪T) + A (
d(S∪T)) for any finite partition.
n n n n
(
d is the partial-derivative symbol.)
+ -
(c) Show that A (S) + A (T) = A(S∪T).
B.
2 2
(a) Find the volume common to the two cylinders x + z ≦1
2 2
and y + z ≦1.
2 2
(b) Find the volume common to the three cylinders x + z ≦1
2 2 2 2
, y + z ≦1 and x + y ≦1.
C.
1 1 (x^2)
(a) Evaluate the integral ∫∫e dx dy.
0 y
1 √(1-z^2) √(1-y^2-z^2) 2 2 2
(b) Evaluate the integral ∫∫ ∫ (x + y + z )xyzdxdydz.
0 0 0
(x+y+z)
D. Evaluate the integral ∫ e dxdydz.
{x^2+y^2+z^2≦1}
Hint. Consider the change variable : x' = (x+y+z)/√3
y' = (x-z)/√2
z' = (x-2y+z)/√6
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