MATH 152, FALL SEMESTER 2009COMMON EXAMINATION I - VERSION BName (print):Signature:Instructor’s name:Section No:INSTRUCTIONS1. In Part 1 (Problems 1–10), mark your responses on your ScanTron form using aNo: 2 pencil. For your o wn record, mark your choices on the exam as well.2. Calculators should not be u sed throughout the examination.3. In Part 2 (Problems 11–16), present your solutions in the space provided. Show allyour work neatly and concisely, and indicate your final an swer clearly. You willbe gra ded, not merely on the final answer, but also on the quality and correctness ofthe work leading up to it.4. Be sure to write your name, section number, and version letter of the examon the Scan Tron form.1Part 1 – M ultiple Choice (50 points)Each question is worth 5 points. Mark your responses on the ScanTron form and on theexam itself .1. Compute the indefinite integralZx3p2 + x4dx.(a)(2 + x4)3/24+ C(b)(2 + x4)3/26+ C(c)2(2 + x4)3/23+ C(d)(2 + x4)−1/28+ C(e)(2 + x4)−1/22+ C2. Compute the indefinite integralZex(1 + ex)10dx.(a) 10(1 + ex)9+ C(b) (1 + ex)11+ C(c)(1 + ex)1111+ C(d) ex(1 + ex)1111+ C(e) ex+ e10x+ C23. Evaluate the definite integralZπ/40sec2θ3 + tan θdθ.(a) ln32(b) lnπ12(c) lnπ4(d) lnπ8(e) ln434. Determine the value of the positi ve number b for which the average value of thefunction f (x) = 2 + 6x on the interval [0, b] is 3.(a) 3(b) 1/3(c)√10 − 13(d) 2(e) 1/65. Compute the indefinite integralZxe−3xdx.(a) −xe−3x3−e−3x9+ C(b) −xe−2x2−e−2x4+ C(c) xe−2x+ e−2x+ C(d) xe−3x+ e−3x+ C(e)x2e−2x2+ C36. Compute the indefinite integralZ2 cos2θ dθ.(a) θ −sin(2θ)2+ C(b) θ + 2 sin(2θ) + C(c)θ2−sin(2θ)4+ C(d)θ2+sin(2θ)4+ C(e) θ +sin(2θ)2+ C7. Calculate the area of the region enclosed by the x-axis, y = ln x, x = e, and x = e3.(a) e2(b)1e3−1e(c)1e2−1e(d) 2e3(e) 3/28. An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work neededto pump half the water out of the a quarium. (The density of water, ρ, is 1000 kg/m3,and acceleration due to gravity, g, is 9.8 m/s2.)(a) 250 J(b) 9.8 × 103J(c) 2.45 J(d) 2.45 × 103J(e) 4.9 J49. Let R denote the region enclosed by the y-axis, the line y = 1, and the curvey =√x. Compute the volume of the solid whose base is R and whose cross sectionsperpendicular to the y-axis are semicircles.(a) π/2(b) π/10(c) π/20(d) π/24(e) π/4010. Suppose that f is continuous on (−∞, ∞), and that F is an antiderivative of f in(−∞, ∞). Which of following is an antiderivative of the function g(x) = f(3x − 2)?(a)F (2x − 3)2(b) F (3x − 2)(c) F (2x − 3)(d)F (3x − 2)3(e) insufficient information to make a determination5Part 2 (56 points)Present your solutions to the following p roblems (11 – 1 6 ) in the space provided. Show allyour work neatly and concisely, and i ndicate your final answer clearly. You will begraded, not merely on the final answer, but also on the quality and correctness of the workleading up to it.11. (10 points) Compute the following integral:Zcos3(√x)√xdx612. (10 points) Compute the following integral:Zsec3x tan3x dx713. (10 points) Let R denote the region bounded by the parabola y = 1 − x2and thestraight line y = −2x − 2. Sketch R and calculat e it s area.814. (10 points) Let R denote the region enclosed (in the first quadrant) by the x-axis,the line x = 1, and the curve y = x3. Sketch R and use the method of cylindricalshells to calculate the volume of the solid obtained by rotating R about the line x = 1.915. (10 points) Let T denote the tri angular regio n with vertices at (0, 0), ( 2 , 1) and (4, 1).Sketch T and use the method of disks to compute the volume of the solid obtainedby rotating T abo ut the y-axis.1016. (6 points) Let f be a function such that f′′is continuous in the interval [0, π]. Giventhatf(0) = −1, f(π) = 1, andZπ0f(x) sin x dx = 4,evaluateZπ0f′′(x) sin x dx.11QN
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