課程名稱︰微積分甲上
課程性質︰大一必修
課程教師︰蔡雅如
開課學院：電資院、管理院、社科院
開課系所︰資工系、資管系、經濟系
考試日期（年月日）︰2014/12/29
考試時限（分鐘）：1HR
試題 :
1. Compute the following integral.
x
(a) ∫─────── dx
√(x^2-4x+13)
π/2 1
(b) ∫ ──────── dx
0 2+sin(x)+cos(x)
∞ x
(c) ∫ ─── dx
1 x^8+4
***[2.][3.][4.(a)(b)]三選二作答***
e^x + e^(-x)
2. Let C be the curve given by f(x)＝cosh(x)＝────── for 0 <= x <= t,
２
where t>0. Show that the arc length of C
is equal to the area bounded by C and the x-axis.
＿＿＿＿
3. Find the centroid of the region bounded by the graphs of y=√r^2-x^2 & y=0.
2 t 2
4. Given a curve C:x(t) = sin (t) , y(t) = 2∫ sin θ dθ , 0 <= t <= π/2 .
0
(a) On what intervals of the curve C is y increasing with respect to x?
(b) On what intervals is the curve C concave upward or concave downward?
***(c)(d)二選一作答***
(c) Find the area enclosed by the curve C, the x-axis and x=1.
(d) Find the exact area of the surface obtained by rotating the curve C about
the y-axis.