課程名稱︰微積分乙下
課程性質︰共同必修
課程教師︰陳其誠
開課學院：管理學院
開課系所︰管院各系、地理系、經濟系
考試日期（年月日）︰2007.06.05
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning.
(１) Calculate the following multiple intergrals:
1 1 1 1
(ａ) ∫ ∫ (x + y) dxdy (ｂ) ∫ ∫ e^(-x^2) dxdy
0 0 0 y
(7 points) (8 points)
(２) Calculate the following double intergrals ∫ ∫f(x, y)dA (7 points each):
R
(ａ) f(x, y) = xy ,
R is the triangle with vertices (-π, 0), (π, 0), (π,π/2)
(ｂ) f(x, y) = xe^x ,
R is the triangle bounded by x + y = 4, x = 0, y = 0
y
(ｃ) f(x, y) = ──── ,
1 + x^2
R is the region bounded by y = 0, y = √x , x = 4
(ｄ) f(x, y) = √(16 - x^2 - y^2) ,
R is the region bounded by x^2 + y^2 = 16
(ｅ) f(x, y) = 2 ,
R is the region bounded by 1 ≦ x + y ≦ 2 , 3 ≦ x - 2y ≦ 4.
(３) Calculate the triple intergral (10 points)
1 √(1 - x^2) √(1 - x^2 - y^2)
∫ ∫ ∫ √(x^2 + y^2 + z^2) dzdydx
-1 -√(1 - x^2) -√(1 - x^2 - y^2)
(４) Calculate the triple intergral (10 points)
∫∫∫ x + y + z dV
Q
where Q is the solid region bounded by 0 ≦ x + y + z ≦ 1 ,
0 ≦ y + z ≦ 1 , 0 ≦ z ≦ 1 .
(５) Let f(x, y) = xy and let D = {(x, y)│x^2 + y^2 ≦ 1}. (25 points)
(ａ) Find critical points of f(x, y) inside the region D. And determine
if every critical point is a local extrema.
(ｂ) Use the Lagrange multiplier method to determine the maximum / minimum
of f(x, y) on the curve x^2 + y^2 = 1 .
(ｃ) Find the maximum / minimum of f(x, y) on D.
(６) Suppose that f(x, y, z) is a continuous function, And assume that
f(0, 0, 0) = 3. For each a, let
a √(a^2 - x^2) √(a^2 - x^2 - y^2)
V(a) = ∫ ∫ ∫ f(x, y, z) dzdydx
-a -√(a^2 - x^2) -√(a^2 - x^2 - y^2)
Explain why
V(a)
lim ──── = 4π (5 points)
a→0 a^3
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◆ From: 118.169.196.178
※ 編輯: YSClaire 來自: 118.169.196.178 (02/20 17:52)