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MATH 172.504 Examination 1 February 14, 2012 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 7 PAGES (INCLUDING THIS ONE). Do not mark in the box below. 1–5 6 7 8 9 Total Points Possible Credit 30 10 24 24 8 96 NAME MATH 172 Examination 1 Page 2 Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer. Z 1. Evaluate 2 √ 6x 2x2 + 1 dx. 0 (A) 26 (B) 27 (C) 28 (D) 29 (E) 30 2. Consider the region in the first quadrant bounded by y = x2 , y = 9, and x = 0. Which integral below gives the volume of the solid obtained by rotating this region about the x-axis? 9 Z (9 − x2 ) dx (A) π 0 3 Z x(9 − x2 ) dx (B) 2π 0 Z (C) 2π 9 (x4 − 9) dx 0 Z 3 (81 − x4 ) dx (D) π 0 Z (E) π 3 y 2 dy 0 February 14, 2012 NAME 3. MATH 172 Examination 1 Page 3 Our ideal spring follows Hooke’s Law. It requires 5 Newtons of force to be held 10 cm from rest. How much work (in Joules) is done in stretching the spring from rest to 20 cm? (A) 0 (B) 1 (C) 10 (D) 100 (E) 1000 Z 4. Let F (x) = x √ t3 + 1 dt. Which of the following statements about F (x) are true? 1 I. F (1) = 0. II. F 0 (2) = 2. III. F 0 (2) = 3. (A) I only (B) II only (C) III only (D) I and II only (E) I and III only February 14, 2012 NAME 5. MATH 172 Examination 1 Page 4 What is the average value of f (x) = cos(3x) on the interval 0 ≤ x ≤ π2 ? (A) − 2 3π (B) − 3 2π (C) 1 3π (D) 1 2 (E) 0 6. [10 points] Let R be the region in √ the first quadrant of the xy-plane bounded above by the union of the curve y = 2 x and the line y = −x + 8 and bounded below by the x-axis. Find the area of R by setting up the integral with respect to y. February 14, 2012 NAME 7. MATH 172 Examination 1 Page 5 [24 points] Evaluate the following integrals. Z (a) π/3 4 cos2 θ sin θ dθ 0 Z √ x2 x + 1 dx Z e2x dx e2x + 1 (b) (c) February 14, 2012 NAME 8. MATH 172 Examination 1 Page 6 [24 points] Let R be the region in the first quadrant of the xy-plane bounded by the curves πx , y = 2 − 2x2 , and the y-axis. y = cos 2 Set up integrals with respect to x (do not evaluate!) that express the volumes of the solids obtained by rotating R about the following axes of rotation. (It may help to start by sketching the region R.) (a) The x-axis. (b) The y-axis. (c) The line y = −1. February 14, 2012 NAME 9. MATH 172 Examination 1 Page 7 [8 points] Let S be the 3-dimensional solid whose base is the region in the first quadrant of the xy-plane bounded by y = cos x, the x-axis, and the y-axis, and whose cross-sections perpendicular to the x-axis are squares. Write down an integral for the volume of S. (You do not need to evaluate the integral.) There is more space on this page than you need for this problem. Feel free to use extra space here and on the back of this page for scratch work. February 14, 2012