MATH 152, Spring 2009COMMON EXAM II - VERSION ALAST NAME, First name (print):INSTRUCTOR:SECTION NUMBER:UIN:SEAT NUMBER:DIRECTIONS:1. The use of a calculator, laptop or computer is prohibited.2. In Part 1 (Problems 1-10), mark the correct choice on your ScanTron using a No. 2 pencil. For your own records,also record your choices on your exam!3. In Part 2 (Problems 11-15), present your solutions in the space provided. Show all your work neatly and conciselyand clearly indicate your final answer. Yo u will be graded not merely on the final answer, but also on the qualityand correctness of the work leading up to it.4. Be sure to write your name, section n umber and version letter of the exam on the ScanTron form.THE AGGIE CODE OF HO NOR“An Aggie does not lie, cheat or steal, or tolerate those who do.”Signature:DO NOT WRITE BELOW!QuestionPoints Awarded Points1-10 4011 1212 1213 1214 1215 121001PART I: Multiple Choice1. (4 pts) Find the length o f the curve y =√x3from the point (0, 0) to the point (4, 8).(a)32(10√10 − 1)(b)49(c)23(10√10 − 1)(d)827(10√10 − 1)(e)232. (4 pts) The improper integralZ∞0xe−3xdx(a) Converges to 9(b) Converges to19(c) Converges to 0(d) Converges to −19(e) Diverges23. (4 pts)Z21x2+ 1x2+ xdx =(a) 1 + 3 ln 2 − 2 ln 3(b) 1 − 3 ln 2 + 2 ln 3(c) 2 + 3 ln 2 − 2 ln 3(d) 2 − 3 ln 2 + 2 ln 3(e) None of the above.4. (4 pts) Find the surface area obtained by rotating the curve y = 4 − x2, 0 ≤ x ≤ 2, a round the y-axis.(a)π6(5√5 − 1)(b)4π3(17√17 − 1)(c)4π3(5√5 − 1)(d)π6(e)π6(17√17 − 1)35. (4 pts) Using the error bound formula |ET| ≤K(b − a)312n2, where K = max|f′′(x)| for a ≤ x ≤ b, what is the smallestvalue of n so that the approximation Tn(The trapezoidal rule with n subintervals) to the integralZ31ln x dx isaccurate to within12400?(a) n = 40(b) n = 2 0(c) n = 60(d) n = 3 0(e) n = 706. (4 pts) A group o f calculus teachers were sitting around an odd shaped pool (see figure below). The widths (inmeters) of this pool were measured at 2-meter interva ls as indicated. Use Simpson’s rule with n = 4 to approximatethe area of this pool.1.5 m2 m3 m(a) 22 square meters(b)223square meters(c)173square meters(d)172square meters(e)443square meters47. (4 pts) The integralZ∞1dxx + e5x(a) Diverges by comparison toZ∞1dxx(b) Converges by comparison toZ∞1dxe5x(c) Diverges by comparison toZ∞1dxe5x(d) Converges by comparison toZ∞1dxx(e) Converges to 0.8. (4 pts) Givendudt= e2t−uand u(0 ) = 1, find u(1).(a) u(1) = ln(12e2+ e −12)(b) u(1) = ln(e + 1)(c) u(1) = ln(2e2+ e − 2)(d) u(1) = ln(12e2− e +12)(e) u(1) = ln(e − 1)59. (4 pts) Find the surface area obtained by rotating the curve x = sin t, y = cos t, 0 ≤ t ≤π3around the x-axis.(a) π(b) π√32(c) π√3(d)π2(e) π√2210. (4 pts) The curve x = e8y, 0 ≤ y ≤ 1 is revolve d around the x-axis. Which of the following integrals gives theresulting surface area?(a)Z102πyr1 +164e16ydy(b)Z102πe8yr1 +164e16ydy(c)Z102πyp1 + 64e16ydy(d)Z102πe8yp1 + 64e16ydy(e)Z102πe8yr1 +164e8y2dy6PART II WORK OUTDirections: Pres e nt your solutions in the space provided. Show all your work neatly and concisely and Box yourfinal answer. You will be gra ded not merely on the final answer, but also on the q uality and cor rectness of the workleading up to it.11. (12 pts) A tank is full of oil and has the shape below. Find the hydrostatic force against one end of the semi-cir culartank. Note the weight density of oil is ρg = 9000 Newton’s per cubic meter.10 m1 msemicircle712. A tank contains 250 liters of pure water. Brine that c ontains 0.01 kg of salt per liter enters the tank at a rate of 20liters per minute. The solution is kept mixed and drains from the tank at a rate of 20 liters per minute. How muchsalt is in the tank after t minutes?813. (12 pts) FindZx + 2x2(x2+ 1)dx914. (12 pts) Find a general solution to the differential equation xdydx= x(ln x)2+ y.1015. (12 pts) Find the centrio d of the region bounded by y =√x and y = x3. Simplify your answer.End of
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