課程名稱︰微積分甲下
課程性質︰必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/05/12
考試時限（分鐘）：40
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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※ 編輯: impin 來自: 140.112.240.100 (06/08 15:38)