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）︰2016/07/21
考試時限（分鐘）：90
試題 :
Part I (30%) Chapter 6.1-6.3
1. (15%) Find the number b such that the line y = b divides the region bounded
by the curves y = x^2 and y = 4 into two regions with equal area.
2. (15%) Find the volume generated by rotating the region bounded by the given
curves about the specified axis.
2a. (5%) Sketch the region which is bounded above by y = 4x - x^2 and below
y = 3. Also, mark the line x = 1 in your graph.
2b. (10%) Find its volume.
Part II (40%) Evaluate the following integrals (Chapter 7.1-7.4)
3. (8%) Determine a and n in the following identity
∫tan(x)^5 dx - ∫tan(x)^3 dx = (tan(x)^n)/a
註:助教表示，如果覺得中間的 - 應該要是 + 的話可以自己改，算得出來就可以
4. (8%) ∫cos(x)^2 tan(x)^3 dx.
5. (8%) ∫ln(3x)^2 dx.
6. (8%) ∫1/((x^2 + 1)(x - 2)) dx.
7. (8%) ∫1/(1+e^x) dx.
Part III Evaluate the following integrals (Chapter 7.5-7.8) (30%)
8. (16%) Find ∫ln(x^2 + x + 1)/x^2 dx
9. (14%) For each of the following integrals, state whether it is convergent
or divergent and give your reasons.
(a) (7%) ∫1 to ∞ x^3/(ln(x) + x^4) dx
(b) (7%) ∫0 to ∞ dx/(x^3 + x^(1/2))