math13 - Math 2153, Exam II, March 24, 2010 Name: Score:...

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Unformatted text preview: Math 2153, Exam II, March 24, 2010 Name: Score: Read the problems carefully before you begin. Show all your work neatly and concisely, and indicate your final answer clearly. Total points = 50. 1. (8 points) Find the centroid of the region bounded by the given curves: y = x 2 , x = y 2 . Solution Notice that the two curves intersect at (0 , 0) and (1 , 1) . For x 1 , we also know that y = x is greater than y = x 2 . Therefore we set f ( x ) = x, g ( x ) = x 2 . Then A = Z 1 ( f ( x )- g ( x )) dx = Z 1 ( x- x 2 ) dx = ( 2 3 x 3 / 2- 1 3 x 3 ) | 1 = 1 3 , and x = 1 A Z 1 x ( f ( x )- g ( x )) dx = 3 Z 1 x ( x- x 2 ) dx = 3( 2 5 x 5 / 2- 1 4 x 4 ) | 1 = 9 20 , y = 1 A Z 1 1 2 ( f 2 ( x )- g 2 ( x )) dx = 3 Z 1 1 2 ( x- x 4 ) dx = 3 2 ( 1 2 x 2- 1 5 x 5 ) | 1 = 9 20 . The centroid is located at ( 9 20 , 9 20 ) . 1 2. (10 points) Find a formula for the general term a n , n = 1 , . . . , of the following sequences, then evaluate the limit of the sequence....
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math13 - Math 2153, Exam II, March 24, 2010 Name: Score:...

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