MA132—Calculus II Name:Exam 2 Student Number:13 June 2006Instructions:• Show your work.• Answers without sufficient justification may not receive full cr edit.• No books, notes, or calculators.• Time limit: 90 minutes.• Some integrals which might help:Zsec(x) dx = ln |sec(x) + tan(x)| + CZcsc(x) dx = ln |csc(x) − cot(x)| + CDo not write below this lineProblem 1: 12 pointsProblem 2: 12 pointsProblem 3: 12 pointsProblem 4: 12 pointsProblem 5: 12 pointsProblem 6: 10 pointsProblem 7: 10 pointsProblem 8: 10 pointsProblem 9: 10 pointsScore = 100 points total(12) Problem 1. Consider the region between the curves y = 2 − x and y = x2for 0 ≤ x ≤ 2.(a) Sketch the region. Label the axis scales and any points of intersection.(b) Write an expression which represents the area of the region. Do not evaluate it.Page 2 of 10(12) Problem 2. Consider the solid obtained by rotating the region bounded by the curves y = x2and y = 4 for 0 ≤ x ≤ 2 about the y-axis.(a) Sketch the region and the solid.(b) Set up an integral for the volume using disks or washers and evaluate the integral.Page 3 of 10(12) Problem 3. Consider the solid obtained by rotating the region bounded by the curves y = 4x−x2and y = −5 about the line x = −2.(a) Sketch the region and the solid.(b) Set up an integral for the volume using cylindrical shells. Do NOT evaluate the integral.Page 4 of 10(12) Problem 4. Solve the initial value problem:dydx= 2xy2+ 2x, y(1) = 0.Page 5 of 10(12) Problem 5. Consider the curve y =√x for 4 ≤ x ≤ 9.(a) Set up an integral for the length of the curve. Do NOT evaluate the integral.(b) Set up an integral for the area of the surface obtained by rotating the curve about the x-axis.Then evaluate the integral. Leave your answer in terms of roots (or fractional powers).Page 6 of 10(10) Problem 6. A 1600-lb. elevator is suspended by a 200-ft. cable that weighs 10 lbs./ft. How muchwork is required to raise the elevator from the basement to the third floor, a distance of 30 ft.?Page 7 of 10(10) Problem 7. Find the average value of the function f (t) = t sin(t) on the interval 0 ≤ t ≤ π.Page 8 of 10(10) Problem 8. A bacteria culture grows with constant relative growth rate. Initially there are 200bacteria, and after 2 hours the count is 1800. Find an expression for the bacteria count as a functionof time t in hours.Page 9 of 10(10) Problem 9. Consider the curve defined by the parametric equationsx = e−t, y = e2t, −1 ≤ t ≤ 1.(a) Eliminate the parameter t to find a Cartesian equation of the curve.(b) Sketch the curve. Label with appropriate values of t and indicate with an arrow the directionin which the curve is traced as t increases.Page 10 of