hw6sol - 3 ). Solution: Here, the center as well as the...

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Fall 2010, MAT21B, Solution to Paper Homework 6 1. By revolving the curve y = x 2 / 3 with 0 x 8 ft along the y - axis, we get a tank. Suppose this tank is filled with seawater whose weight-density is 64 lb/ft 3 . How much work does it take to pump the liquid to the level of the top of the tank? Solution: W = X k W k = X k 64 π ( x ) 2 Δ y k (4 - y ) = X k 64 πy 3 Δ y k (4 - y ) Z 4 0 64 πy 3 (4 - y ) dy = 64 π ± y 4 - y 5 5 ²³ ³ ³ ³ 4 0 = 2 14 π 5 ft-lbs 2. A circular plate with diameter 4ft is submerged vertically in a swimming pool. The distance from the top of the plate to the water surface is 3 ft. Find the force exerted by the water against one side of the disk. (Water’s weight-density is 62.4 lb/ft
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Unformatted text preview: 3 ). Solution: Here, the center as well as the area of disk is clear, so we use the formula F = w hA to calculate the force. F = w hA = 62 . 4 (3 + 2) 2 2 = 62 . 4 20 = 1248 lbs. 3. Evaluate the integral Z 8 1 log 8 x x dx Solution: Let u = log x 8 , then du = dx x ln8 . Also, when x = 1, log x 8 = log 1 8 = 0, and when x = 8, log x 8 = log 8 8 = 1 . So, Z 8 1 log 8 x x dx = Z 1 u ln 8 dx = u 2 2 ln 8 | u =1 u =0 = ln 8 2 = ln 2 3 2 = 3 2 ln 2 . 1...
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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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