課程名稱︰微積分甲上
課程性質︰必修
課程教師︰李白飛
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰101/1/10
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Let f(x) be a continuous function on the positive real axis such that
xy x y
∫f(t)dt = y∫f(t)dt + x∫f(t)dt for all x>0 and all y>0. If f(1)=3, compute
1 1 1
f(x) for each x>0. (Simplify your answer!)
2.Show that the function f(x) = x + e^x has an inverse function on R. Let g(x)
be the inverse function of f(x). Find the area of the region bounded bt y=g(x)
and the lines y=0 and x=1+e.
x ln(t)
3.Let f(x) = ∫ -------dt for x>0. Compute f(x)+f(1/x). (Simplify your answer!)
1 (t+1)
1
4.Obtain and use a reduction formula to evaluate I_n=∫(1-x^2)^n dx. (n=1,2...)
0
e^(2x)
5.Evalute ∫-------------dx.
(e^x+1)^(1/4)
(pi/2) sin(x)
6.Evalute ∫ ---------------dx.
0 1+cos(x)+sin(x)
1
7.Evalute lim ---[√(n^2+1^2)+√(n^2+2^2)+...+√(n^2+n^2)]
n→∞ n^2
8.Evalute lim x[(1+1/x)^x-e].
n→∞
9.A drinking glass is a right-circular cylinder of radius a and height h.
It is tilted until the water level bisects the base and touch the rim.
Find the volume of the remained water.
(很難理解的題目，就是一杯水倒到杯底半圓，問剩下水的體積)
10.Find the centroid of a sector of radius a and angular width θ.
然後這張好像很難，平均不及格
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