Math 132 Fall 2007 Exam III 1. The region in the first quadrant that is bounded above by = = = = y − − − − 4 x x2 and bounded below by = = = = y x is rotated about the vertical line = = = = x −−−−1. What is the volume of the resulting solid of revolution?a) 15 ππππ2 b) 16 ππππ c) 18 ππππ d) 21 ππππ e) 45 ππππ2 f) 20 ππππ g) 27 ππππ2 h) 25 ππππ2 i) 14 ππππ j) 15 ππππ2. The region in the first quadrant bounded above by = = = = y x and below by = = = = y x2 for ≤ ≤ ≤ ≤ 0 x ≤ ≤ ≤ ≤ 1 is rotated about the line y = 2. What is the volume of the solid of revolution that results?a) 8 ππππ15 b) 3 ππππ5 c) 2 ππππ3 d) 4 ππππ5 e) 14 ππππ15 f) ππππ2 g) 3 ππππ8 h) 5 ππππ8 i) 3 ππππ4 j) 7 ππππ83. Calculate the arc length of the graph of = = = = y + + + + x3314 x for ≤ ≤ ≤ ≤ 12x ≤ ≤ ≤ ≤ 1. a) 512 b) 1124 c) 12 d) 1324 e) 712 f) 58 g) 23 h) 1724 i) 34 j) 564. The line segment with end points (2,3) and (6,0) is rotated about the y-axis. What is the surface area of the resulting conical frustum?a) 12 ππππ b) 16 ππππ c) 20 ππππ d) 24 ππππ e) 25 ππππ f) 27 ππππ g) 28 ππππ h) 32 ππππ i) 36 ππππ j) 40 ππππ5. The graph of = = = = y 3 x for ≤ ≤ ≤ ≤ 4 x ≤ ≤ ≤ ≤ 10 is rotated about the x-axis. What is the surface area of the resulting figure? a) 54 π b) 60 π c) 64 π d) 72 π e) 80 π f) 84 π g) 94 π h) 102 π i) 109 π j) 115 π6. Let R be the trapezoidal region in the first quadrant that is bounded above by the graph of = = = = y 2 x, below by the x-axis, on the left by the vertical line = = = = x 1, and on the right by the vertical line = = = = x 2. What is the moment of R about the vertical line = = = = x −−−−1 ? (Assume that R has a constant mass density equal to 1.)a) 203 b) 7 c) 233 d) 8 e) 253 f) 9 g) 323 h) 11 i) 353 j) 127. Let R be the region in the first quadrant that is bounded above by the graph of = = = = y − − − − 4 x2 and below by the x-axis. What is the x-coordinate of the center of mass of R? a) 13 b) 38 c) 25 d) 12 e) 35 f) 58 g) 23 h) 34 i) 54 j) 328. What is the y-coordinate of the center of mass of the region R of the preceding question? a) 53 b) 65 c) 118 d) 43 e) 75 f) 138 g) 85 h) 74 i) 95 j) 1589. Suppose that = = = = ( )f x x2. If ( )f 7 is equal to the average value of ( )f x over the interval [ ],2 b , then what is b ? a) 8 b) 253 c) 9 d) 293 e) 10 f) 313 g) 11 h) 353 i) 12 j) 37310. Suppose that = = = = ( )fxc + + + + 1 x2 is the probability density function of a random variable that has values in the interval [ ],0 3 . What is c ? a) 1ππππ b) 2ππππ c) 3ππππ d) 4ππππ e) 5ππππ f) ππππ g) ππππ2 h) ππππ3 i) ππππ4 j) ππππ511. A random variable X that assumes values in the interval [ ],0 1 has probability density function = = = = ( )f x8( ) + + + + ππππ2 ( ) + + + + 1 x22 for ≤ ≤ ≤ ≤ 0 x ≤ ≤ ≤ ≤ 1. If µµµµX is the mean of X, what is ( ) + + + + ππππ2µµµµX ? a) 2 b) 2 c) 3 d) 3 e) 5 f) 5 g) 6 h) 6 i) 2 2 j) 2 312. A random real number X is chosen from the interval [ 1 , 9 ]. If the p.d.f. of X is = = = = ( )f x3 x52, then what is the probability that X is greater than 4 ? a) 713 b) 1526 c) 813 d) 1726 e) 913 f) 1926 g) 1013 h) 2126 i) 1113 j) 232613. A random variable X has pdf given by = = = = ( )fx + + + + 2 x 1108 for ≤ ≤ ≤ ≤ 1x ≤ ≤ ≤ ≤ 10. What is the median m of X? a) 5 b) 16/3 c) 17/3 d) 6 e) 13/2 f) 7 g) 22/3 h) 23/3 i) 8 j) 17/214. Starting from equilibrium, 60 J of work are expended when a spring is stretched 2 meters. How many meters beyond equilibrium can the spring be stretched with a force of 45 N? a) 15 b) 12 c) 1 d) 65 e) 32 f) 85 g) 2 h) 125 i) 52 j) 315. The vertical cross-sections of a tank have the shape = = = = y − − − − x 12 for ≤ ≤ ≤ ≤ −−−−12 x ≤ ≤ ≤ ≤ 12 (measured in meters). Horizontal cross-sections of the tank are rectangles that are 5 m long. The widths of these rectangles vary, but can be deduced from the description of the vertical cross-sections. The tank is filled to the top with a fluid that has weight density equal to 310 Nm3. If after pumping water to the top of the tank, the remaining water is 10 m deep, how many Joules of work have been done?a) 24 b) 32 c) 36 d) 48 e) 52 f) 56 g) 64 h) 72 i) 80 j) 8416. Forty feet of a uniform cable hang over the side of a building. The cable weighs 2 lbs/ft. A 48 pound hook is attached to the dangling end of the cable. The cable is pulled up 30 feet before the hook snags on a balcony rail. How many foot-pounds of work have been done?a) 2480 b) 2540 c) 2580 d)
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