[試題] 102上 張志中 微積分甲上 第四次小考
課程名稱︰微積分甲上
課程性質︰必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰102/11/26
考試時限（分鐘）：30分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Calculus A2 (102 Fall) Quiz Four
17:50-18:20, November 26, 2013
It's necessary to explain all the reasons in detail and show all of your
work on the answer sheet. Or you will NOT get any credits. If you used any
theorems in textbook or proved in class, state it carefully and explicitly.
1. Evaluate the integral.
3 8
(a) (15pts) ∫sin xcos xdx
2
(b) (15pts) ∫xcos xdx
2
3 -x
(c) (15pts) ∫x e dx
-1 2
(d) (15pts) ∫(sin x) dx
2 3
2. Let R be the rigion bounded by the curves y=3x+2x -x and y=3x.
Calculate the following quantities.
(a) (10pts) The area of R.
(b) (15pts) The volume obtained by rotating R about the y-axis.
3. (15pts) Use integration by parts to prove the reduction formula.
n-2
n tanxsec x n-2 n-2
∫sec xdx=———————+————∫sec xdx
n-1 n-1
Answer:
9 11
cos x cos x
1. (a) -———+——— +C
9 11
2
x xsin2x cos2x
(b) ——+————+——— +C
4 4 8
2 2
1 2 -x 1 -x
(c) -—x e -—e +C
2 2
-1 2 2 -1
(d) (xsin x) +2√(1-x )sin x-2x+C
2. (a) 4/3
(b) 16π/5
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※ 編輯: h2s4 來自: 180.177.124.246 (01/17 23:28)