課程名稱︰微積分甲上
課程性質︰必修
課程教師︰顏文明(陳宏代課)
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/07/28
考試時限（分鐘）：90
試題 :
Part I (40%) Chapter 8.1-8.2
1. (16%) Find the length of the arc of semicubical parabola y^2 = x between
the points (0,0) to (1,1).
(1a). (8%) Note that its length can be written of the form
∫_a^b (1+cy^d)^(1/2) dy. Please give a, b, c, and d.
(1b). (8%) Pleae give the length. (You need to show the process of getting
your answer.)
2. (14%) Find the area of the surface generated by revolving the curve
y = (4 - x^2)^(1/2) on the interval [0,1], about the y-axis
3. (10%) Evaluate the area of the surface generated by revolving the curve
y = x^3/3 + 1/(4x), 1 ≦x ≦3, about the line y = -2
Part II (20%) Chapter 9.1, 9.3
4. (10%) Find the solution of dy/dx = xe^(-x) and y(0) = 1
5. (10%) Find the solution of dy/dx = y(1 - lny) and y(0) = y_0
Part III (20%) Evaluate the following improper integrals (Chapter 7.8)
6. (12%) ∫_(-∞)^(∞) 1/(e^x + e^(-x)) dx
7. (8%) Does the integral ∫_2^(∞) (2 + sinx)/(x-1) dx converge?
Part IV (20%) Evaluate the following integrals
8. (10%) Evaluate ∫dx/(16+9x^2)^(1/2) dx.
9. (10%) Evaluate ∫sin(ln(t))dt. (Hint: Consider making the substitution
t = e^x.)