Math 132 Spring 2011 Exam 2 NAME STUDENT ID NUMBER This exam contains sixteen questions The first fourteen are multiple choice questions and count for five points each There is no partial credit on these questions so read each question carefully check your arithmetic and make sure that you have marked the answer you intended to mark The last two questions which are each worth fifteen points require written answers and some partial credit might be given However no credit will be given for information that is not germane to the problem at hand Please make sure to write your name and student ID number on the pages that include your answers to the last two questions In fact you will get one point on each of these two questions for writing your name and ID number legibly 1 1 Use the Trapezoidal Rule with n 4 to approximate the integral Z 5 1 dx 1 x a ln 5 b 5 3 c 11 6 d 19 7 e 101 60 f 103 30 g 1 6094 h 2 3667 2 2 Suppose that on the interval 0 10 the function f is continuous f x 0 f is decreasing and f has graph which is concave up For any value of n list the numbers Ln Rn Tn E in increasing order where Ln is the left endpoint approximation Rn is the right endpoint approximation Tn is the trapezoidal approximation and E is the exact value of the integral Z 5 1 f x dx a Rn E Ln Tn b Rn E Tn Ln c Rn Tn E Ln d Rn Tn Ln E e Ln E Rn Tn f Ln E Tn Rn g Ln Tn E Rn h Ln Tn Rn E 3 3 How large do we have to choose n so that the error in using Simpson s Rule to approximate Z 17 1 is less than 10 4 a 1 b 2 c 16 d 24 e 134 f 472 g 26 368 h 46 668 4 1 dx x4 4 Compute Z e 1 dx x lnx 3 a diverges to b diverges to c diverges but not to or d converges to e e converges to e f converges to 1 g converges to e 1 h converges to 1 2 5 5 Find the area of the region bounded by the curves y ex y x2 1 x 1 and x 1 a e 1 e 4 3 b e 1 e 1 3 c e d 1 e e e 1 e f e 1 e g 1 e h 2 e 6 6 Find the area of the region bounded by the curves x 1 y 2 and x y 2 1 a 2 b 0 c 2 2y 2 d 1 2 e 1 f 4 3 g 8 3 h 15 4 7 7 Compute the volume of the solid obtained by rotating the region in the first quadrant bounded by y x2 y 4 and x 0 about the y axis a 40 2 3 b 8 c 32 5 d 16 3 e 16 3 f 16 2 3 g 64 3 h 12 8 8 Compute the volume of the solid obtained by rotating the region in the first quadrant bounded by y x2 y 2 x2 and x 0 about the x axis a 11 5 b 8 3 c 32 15 d 16 5 e 16 3 f 16 2 3 g 64 15 h 12 5 9 9 Compute the volume of the solid obtained by rotating about the line x 1 the region bounded by y sin x y 0 0 x 2 a 1 b c 2 d 4 e 1 2 f 3 2 g 3 h 2 3 10 10 Find the exact length of the curve given parametrically by x 1 3t2 y 4 t3 where 0 t 5 a 1 b 19 c 25 d 45 e 3 5 f 5 5 g 6 5 h 9 5 11 3 11 Find the exact length of the curve given by x 23 y 2 0 y 1 a tan 1 3 2 b tan 1 7 c d e f g h 4 2 3 4 2 1 3 9 2 4 4 2 27 4 2 2 3 3 2 2 12 12 Find the average value of f x e e2 a 1 e b 1 e 1 1 e 1 1 e2 e 2 e2 e ln 2 e2 e ln ln2 e2 e ln2 2 e2 e c d e f g h 13 1 xln x on the interval 13 Let f x c 1 x2 For what values of c is f a probability density function a no value of c makes f a probability density function b 0 c 1 d 2 e 2 f 2 g 2 h 1 14 14 Suppose you are given the probability density function xe x x 0 f x 0 x 0 Find P 1 X 2 a e2 e 2 b c d e f g h 2 3 e e2 1 2 e2 e 2 1 e2 e 2 3 e2 e 3 2 e e2 1 1 e e 1 1 1 e2 e 15 Name Student ID 15 Find the volume of the solid generated by revolving the region bounded by y 1 x2 and the x axis about the line x 4 a Use Method I disc washer method b Use Method II cylindrical shells method 16 Name 16 Evaluate Student ID Z 2 0 ln x dx Show all work 17 Name Student ID 18 Name Student ID 19
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