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Math 152, Summer2008-copyright Joe Kahlig Page 1 For questions 1-4 you do not have to compute out the integral(s) to a final number. Also, do not use the absolute value symbol in any of the integrals. √ 1. (7 points) The region bounded by the curves y = sin(x2 ) and y = 0 on the interval x = 0 to x = π is shown in the picture. Give the integral(s) that would compute the volume when the region is rotated about x = −2. 2. (8 points) Sketch the graph of the region and set up the integral(s) that would give the area of the region that is bounded by the functions x = 5y − y 2 and x = 6 − 2y from the x-axis to y = 4. 3. (7 points) A solid has a base that is bounded by y = x4 and y = 16. Parallel cross sections are perpendicular to the x−axis are rectangles whose height is twice the length of the base. Note: consider the base of the rectangle to be also the base of the solid. Give the integral(s) that would compute the volume of the solid. 4. (7 points) A solid is obtained by rotating the region bounded by the given curves about the x-axis. Give the integral(s) that would compute the volume. √ y =2 x−2 x=y+2 5. (7 points) A spring has a natural length of 2ft. If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch the spring 9 inches beyond its natural length? 6. (10 points) A tank has the shape of an inverted circular cone with a height of 8m and a base radius of 3m. The tank is filled with water to a height of 6m. Set up the integral(s) that would give the work required to empty the tank by pumping all of the water to the top of the tank. The weight density of water is δ = ρg = 62.5 lb/ft3 or δ = ρg = 9800 N/m3 . Note: Be sure that you indicate where you are placing the axis and what direction is positive. 7. (8 points) Find the average value of the function f (x) = (x2 + 1)ex on the interval [1, 4]. 8. (6 points) Complete the square: 12x − 3x2 + 5 = 9. (8 points each) Evaluate these integrals. Z (a) sin(2x)ex dx Z (b) tan3 (4x) dx Z (c) cot4 (x) csc4 (x) dx Z √ 2 x −4 (d) dx x4 Z x2 √ (e) dx 9 − 4x2 Check the back of the page for more problems.