作者h2s4 (a8319)
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标题[试题] 102上 张志中 微积分甲上 第四次小考
时间Mon Jan 13 21:36:00 2014
课程名称︰微积分甲上
课程性质︰必修
课程教师︰张志中
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰102/11/26
考试时限（分钟）：30分钟
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
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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