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MATH 172.503 Examination 1 February 18, 2010 NAME SIGNATURE This exam consists of 9 problems, numbered 1–9. For partial credit you must present your work clearly and understandably and justify your answers. The use of calculators is not permitted on this exam. The point value for each question is shown next to each question. CHECK THIS EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE 9 PROBLEMS ON 6 PAGES (INCLUDING THIS ONE). Do not mark in the box below. 1–5 6 7 8 9 Total Points Possible 35 12 25 12 16 100 Credit NAME MATH 172.503 Examination 1 Page 2 Multiple Choice: [7 points each] In each of Problems 1–5, circle the best answer. 1. Evaluate Z π 4 0 (A) cos(2x) dx. 1 + sin(2x) 1 ln(2) 2 (B) ln(2) (C) 1 2 (D) 1 (E) 0 2. Evaluate Z 0 √ 3 √ 5x dx. x2 + 1 (A) 1 (B) 1 3 (C) 35 3 (D) 5 (E) 10 February 18, 2010 NAME 3. MATH 172.503 Examination 1 Page 3 If a spring is pulled far enough, it can become distorted and no longer exhibit the properties of a perfect spring. Suppose that pulling a particular spring a distance x ≥ 1 meters requires a force (in Newtons) of F (x) = 2 ln x x to have the spring remain stretched. How much work (in Joules) is done in pulling the spring from 2 m to 3 m from rest? (A) ln(2) (B) ln(3) (C) ln(3) − ln(2) (D) ln(9) − ln(4) (E) (ln(3))2 − (ln(2))2 4. What is the average value of the function f (x) = cos3 (x) on the interval 0 ≤ x ≤ π2 ? (A) 1 π (B) 2 π (C) 4 3π (D) 1 2 (E) 2 3 February 18, 2010 NAME 5. MATH 172.503 Let F (x) = Z 3 2x √ Examination 1 Page 4 1 dt. What is F ′ ( 12 )? 2 t + 3t (A) 0 (B) 1 (C) 2 6. (D) 1 2 (E) 2 3 [12 points] Find the area of the region in the plane which is bounded by the curves x = 1 − y 2 and x = y 4 − y 2 − 15. February 18, 2010 NAME 7. MATH 172.503 Examination 1 Page 5 [25 points] Evaluate the following indefinite integrals. (a) Z (b) Z x cos(3x) dx (c) Z sin2 (x) dx x5 dx x3 − 1 February 18, 2010 NAME MATH 172.503 Examination 1 Page 6 8. [12 points] Find the volume obtained by rotating the region which is bounded above by y = 4 − x2 and below by the x-axis about the line y = −1. (You do not need to simplify your numerical answer completely.) 9. [16 points] Let A be the first point to the right of the origin where the curves y = sin(2x) and y = sin(x) intersect. Consider the region bounded by these two curves between the origin and A. Set up integrals with respect to x (do not evaluate!) that express the volumes obtained by rotating this region about the following axes of rotation. (a) The x-axis. (b) The y-axis. February 18, 2010