MATH 152, SPRING SEMESTER 2005COMMON EXAMINATION I - VERSION AName (print):Signature:Instructor’s name:Section No:Seat No:INSTRUCTIONS1. Calculators may notbe used on this exam. The ScanTrons will be collected after 90 minutes.2. In Part 1 (Problems 1–10), mark the correct choice on your ScanTron form using a No.2pencil. For your own record, mark your choices on the exam itself, as the ScanTrons will notbe returned.3. In Part 2 (Problems 11–15), present your solutions in the space provided. Show all yourwork neatly and concisely, and indicate your final answer clearly. You will be graded,not merely on the final answer, but also on the quality and correctness of the work leading upto it.4. Be sure to write your name, section number, and version letter of the exam on theScanTron form.QN PTS1–101112131415TOTAL1Part 1 – Multiple Choice (50 points)Read each question carefully; each problem is worth 5points.1. Find the average value of f (x)=xsin(x2) over the interval [0,√π].(a)12√π(b)1√π(c) 1(d) 2(e) 02. The force required to maintain a spring stretched 2 inches beyond its natural length is 3 pounds.How much work is done in stretching the spring from its natural length to 2 feet beyond itsnatural length? Note: 1 foot=12 inches.(a) 3 foot-pounds(b) 18 foot-pounds(c) 12 foot-pounds(d) 36 foot-pounds(e) 42 foot-pounds3. Find the area of the region bounded by y = |x2− 1|, y =0,x=0andx=2.(a) 2(b) 1(c)23(d)32(e)1224. EvaluateZπ20sin2x cos2xdx.(a)π4(b)π8(c)π2(d) π(e)π165. The region bounded by y = ex, y =0,x=0,andx= 3 is revolved around the x-axis. Findthe volume of the resulting solid.(a) 2π(e6− 1)(b) π(e6− 1)(c)π2(e6− 1)(d)π2(e3− 1)(e)π6(e9− 1)6. EvaluateZ31x ln xdx.(a)92ln 3 − 2(b) 3 ln 3 − 2(c)92ln 3(d) ln 3(e) ln 3 −1237.Zsin3x cos2xdx =(a)cos3x3−cos5x5+ C(b)cos5x5−cos3x3+ C(c)sin4x cos3x12+ C(d) −cos3x sin2x + C(e)sin4x4−sin6x6+ C8.Z21xx +1dx =(a) 1 + ln 2(b) 1 + ln 6(c)16(d) 1 − ln23(e) 1 + ln239. The region bounded by y = x2and y =2xis rotated about the x-axis. Find the volume of theresulting solid.(a)43(b)8π15(c)4π3(d)64π15(e)16π15410. Suppose it is given that f (5) = 3, f0(5) = 2, f(1) = 4 and f0(1) = −1. ComputeZ51xf00(x) dx.Hint: Use integration by parts.(a)752(b) 11(c) 12(d) 9(e) Not enough information to determine.5Part 2 (56 points)The use of a calculator is NOT permitted for this part of the exam. All work must be showninorder to receive credit. Refer to the front page for further instructions.11. (10 points) A tank is full of water. Find the work required to pump all of the water to thetop of the tank. Note: The weight density of water is δ = ρg = 9800 Newtons per cubic meter.3 m1 msemicircle12. (10 points) Find the volume of the solid S described below:The base of S is the ellipse x2+y24= 1. Cross-sections perpendicular to the y-axis are squares.613. (12 points) Consider the area of the region bounded by y =√x, y =2−xand y =0.a.) Sketch the bounded region.−3−2 −101 23−3−2−10123b.) Find the volume obtained by revolving the region about the x-axis.714. (12 points)EvaluateZdxx2√25 − x2.815. (12 points)EvaluateZ10(x −
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