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.) --

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