※ [本文轉錄自 NTU-Exam 看板]
作者: Aiwai (亞麻色の髮の乙女(B) 看板: NTU-Exam
標題: [試題] 93下 黃漢水 微積分甲下 期末考
時間: Sun Jun 26 14:49:36 2005
課程名稱︰微積分甲下
課程性質︰必修
課程教師︰黃漢水
開課系所︰數學系
考試時間︰2005.6.24 10：20 ～ 12：10
試題 :
一 Let D be the surface x^2 + 4y^2 + 4z^2 = 16 and the temperature at the
point (x,y,z) on D is T(x,y,z) = 8y^2 + 4xy - 16z + 400. Find the higest
and lowest temperatures on the surface. (20％)
4 1 2 2cos(x^2) ꄊ二 Find the integral ∫｛∫ [∫ ────── dx] dy｝dz (15％)
1 ꈠ0 2y √z ꌊ
三 Let D be a thin plate outside the circle r = 5 and inside the circle
r = 10cosθ. Suppose the density at the point (x,y) is den(x,y) = 1/r .
Find the mass of D. (15％)
四 Let D be region bounded by the cylinder x^2 + y^2 = 9 and the planes z = 0
, y + z = 6. Suppose the density at the point (x,y,z) on D is
den(x,y,z) = 2z. (20％)
(1) Find the mass of D.
(2) Find the center of mass of D.
→ → → →
五 Let F (x,y,z) = 2z i + 3x j + 5y k and S be the surface in the outward
→ → → →
direction r (r,θ) = (rcosθ) i + (rsinθ) j + (9-r^2) k , 0≦ r ≦3,
→ →
0≦θ≦2π. Find ∫∫▽ ×F ‧n dσ. (15％)
s
→ → → →
六 Let F (x,y,z) = 0 i + 0 j + (7z-2) k and Let D =｛(x,y)│x^2 + y^2 ≦9｝.
Suppose f(x,y)≧0 for any (x,y)εD and
K =｛(x,y,z)│x^2 + y^2 ≦9, 0≦ z ≦f(x,y)｝ If the volume of K is 70,
→ →
find the flux ∫∫F‧n dσ. (15％)
s
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※ 編輯: Aiwai 來自: 140.112.249.235 (06/26 17:05)
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