课程名称︰微积分甲上
课程性质︰必修
课程教师︰颜文明(陈宏代课)
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/07/21
考试时限（分钟）：90
试题 :
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
注:助教表示，如果觉得中间的 - 应该要是 + 的话可以自己改，算得出来就可以
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))