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)) --

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