Math 132 Fall 2007 Exam III 1 The region in the first quadrant that is bounded above by y 4 x x 2 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 2 f 20 b 16 g 27 2 45 2 c 18 d 21 e 25 2 i 14 j 15 h 2 The region in the first quadrant bounded above by y x 0 x 1 is rotated about the line y 2 What is the volume of the solid of revolution that results 8 15 f 2 a 3 5 3 g 8 b 2 3 5 h 8 c 4 5 3 i 4 d 14 15 7 j 8 e and below by y x2 for 3 x 1 1 3 Calculate the arc length of the graph of y for x 1 3 4x 2 5 11 1 13 7 a b c d e 12 24 2 24 12 f 5 8 g 2 3 h 17 24 i 3 4 j 5 6 4 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 f 27 b 16 g 28 c 20 h 32 d 24 i 36 e 25 j 40 5 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 f 84 b 60 g 94 c 64 h 102 d 72 i 109 e 80 j 115 6 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 20 3 b 7 f 9 g 32 3 23 3 d 8 h 11 i 35 3 c e 25 3 j 12 7 Let R be the region in the first quadrant that is bounded above by the graph of y 4 x and below by the x axis What is the x coordinate of the center of mass of R 1 3 5 f 8 a 3 8 2 g 3 b 2 5 3 h 4 c 1 2 5 i 4 d 3 5 3 j 2 e 2 8 What is the y coordinate of the center of mass of the region R of the preceding question 5 3 13 f 8 a 6 5 8 g 5 b 11 8 7 h 4 c 4 3 9 i 5 d 7 5 15 j 8 e 2 9 Suppose that f x x 2 b If f 7 is equal to the average value of f x over the interval then what is b a 8 f 31 3 b 25 3 g 11 c 9 h 35 3 d 29 3 i 12 e 10 j 37 3 10 Suppose that f x c 2 is the probability density function of a random variable 1 x that has values in the interval 0 3 What is c a 1 f 2 g 2 b 3 h 3 c 4 i 4 d 5 j 5 e 11 A random variable X that assumes values in the interval 0 1 has probability density function 8 f x 2 2 1 x If for 0 x 1 is the mean of X what is 2 X X a 2 f 2 b 5 2 g 6 c 3 h d 6 3 i 2 2 e 5 j 2 3 12 A random real number X is chosen from the interval 1 9 If the p d f of X is f x 7 13 19 f 26 a 3 x then what is the probability that X is greater than 4 52 15 26 10 g 13 b 8 13 21 h 26 c 17 26 11 i 13 d 9 13 23 j 26 e 13 A random variable X has pdf given by f x 2x 1 for 1 x 10 108 What is the median m of X a 5 f 7 b 16 3 g 22 3 c 17 3 h 23 3 d 6 i 8 e 13 2 j 17 2 14 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 1 5 8 f 5 a b 1 2 g 2 c 1 h 12 5 6 5 5 i 2 d e 3 2 j 3 15 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 3 cross sections The tank is filled to the top with a fluid that has weight density equal to 10 N If after pumping water to the top of the tank the remaining water is 10 m deep how 3 m many Joules of work have been done a 24 f 56 b 32 g 64 c 36 h 72 d 48 i 80 e 52 j 84 16 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 f 2740 b 2540 g 2800 c 2580 h 2840 d 2640 i 2880 e 2680 j 2940 8 17 Calculate a 4 3 f 5 2 1 dx 1 3 8 x 7 b 3 2 g 8 3 c 5 3 h 3 d 2 i 10 3 e 7 3 j 7 2 2 x 18 Calculate dx 2 2 x3 1 a 1 f 1 6 1 2 1 g 7 b 1 3 1 h 8 c 1 4 1 i 9 d 1 5 1 j 10 e th 19 Calculate the 100 partial sum of n 1 n 1 n 1 a 1 f 49 50 99 100 50 g 49 b 100 99 98 h 101 c 100 101 101 i 98 d 101 100 1 j 10100 e 20 Consider the series 2n I n 1 n II 1 tan n n 1 III 1 sec n n 1 IV 1 n 1 n List all given series for which the Divergence Test yields a conclusion a I b II c III f I III g I …
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