hw5sol - x = p 2 y-1 5 8 ≤ y ≤ 1 about the y-axis...

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Fall 2010, MAT21B, Solution to Written Homework 5 1. Find the length of the parametric curve x = e t - t,y = 4 e t 2 , 0 t 1 . Solution: The length of this parametric curve is given by L = Z 1 0 s ± dx dt ² 2 + ± dy dt ² 2 dt = Z 1 0 q ( e t - 1) 2 + (4 e t 2 ) 2 dt = Z 1 0 e 2 t + 2 e t + 1 dt = Z 1 0 p ( e t + 1) 2 dt = Z 1 0 e t + 1 dt = e t + t ³ ³ 1 0 = e 2. Find the centroid (¯ x, ¯ y ) of the region bounded above by the curve y = x +2 and below by y = x 2 . Solution: Recall that ¯ x = M y M = R b a ˜ xdm R b a dm ¯ y = M x M = R b a ˜ ydm R b a dm So, we need to find ˜ x, ˜ y and dm for a typical strip. For a typical vertical strip, we have center of mass (˜ x, ˜ y ) = ´ x, x +2+ x 2 2 µ and dm = dA = ( x + 2 - x 2 ) dx These two curves intersect at ( - 1 , 1) and (2 , 4). Thus, M = Z 2 - 1 dm = Z 2 - 1 ( x + 2 - x 2 ) dx = 9 2 . 1
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M x = Z ˜ ydm = Z 2 - 1 ( x + 2 + x 2 ) 2 ( x + 2 - x 2 ) dx = 1 2 Z 2 - 1 (( x + 2) 2 - ( x 2 ) 2 ) dx = 1 2 Z 2 - 1 ( x 2 + 4 x + 4 - x 4 ) dx = 36 5 and M y = Z ˜ xdm = Z 2 - 1 x ( x + 2 - x 2 ) dx = 9 4 . Therefore, ¯ x = M y M = 9 4 9 2 = 1 2 . ¯ y = M x M = 36 5 9 2 = 8 5 . 3. Find the area of surface generated by revolving the curve
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Unformatted text preview: x = p 2 y-1 , 5 8 ≤ y ≤ 1 about the y-axis. Solution: We have the following information about the curve: x = p 2 y-1 dx dy = 1 √ 2 y-1 Then the area of the surface of revolution, A , is A = Z 1 5 8 2 πx s 1 + ± dx dy ² 2 dy = Z 1 5 8 2 π p 2 y-1 r 1 + 1 2 y-1 dy = Z 1 5 8 2 π s (2 y-1) ± 1 + 1 2 y-1 ² dy = Z 1 5 8 2 π p (2 y-1) + 1 dy = 2 π Z 1 5 8 p 2 y dy = 4 √ 2 π 3 y 3 2 ³ ³ ³ 1 5 8 = 4 √ 2 π 3-5 √ 5 π 12 . 2...
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This note was uploaded on 10/08/2010 for the course MATH 21B taught by Professor Vershynin during the Spring '08 term at UC Davis.

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hw5sol - x = p 2 y-1 5 8 ≤ y ≤ 1 about the y-axis...

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