Use the method of cylindrical shells to find the volume V generated by rotating

Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis. y = 4e^x, y = 4e^?x, x = 1; about the y-axis

Explanation / Answer

using the Washer Method: about y-axis y = 4e^(x). . . .x = ln( y/4 ) y = 4e^(-x). . . .x = - ln( y/4 ) limits: when x = 0 ------> y = 4 when x = 1 ------> y = 4e and 4/e A (y) = p ( outer radius )^2 - p ( inner radius )^2 A (y) = p ( 1 - 0 )^2 - p ( -ln(y/4) - 0 )^2 A (y) = p (1 - ( ln(y/4) )^2 ) A (y) = p ( outer radius )^2 - p ( inner radius )^2 A (y) = p ( 1 - 0 )^2 - p ( ln(y/4) - 0 )^2 A (y) = p (1 - ( ln(y/4) )^2 ) 4. . . .. . . . . . . .. . . .. . 4e ? p (1 - ( ln(y/4) )^2 ) dy + ? p (1 - ( ln(y/4) )^2 ) dy = 16p/e 4/e . .. . . . .. . . . . . . .. . 4 Using shell method: height ===> y = ( 4e^(x) - 4e^(-x) ) radius ====> x limits: 1 ? 2p * x * ( 4e^(x) - 4e^(-x) ) dx 0 1 ? 2p * ( 4xe^(x) - 4xe^(-x) ) dx ---> by parts 0 u = 4x. . . .dv = e^(x) du = 4. . . .v = e^(x) u * v - ? v * du 4x * e^(x) - ? 4 * e^(x) dx 4x * e^(x) - 4 * e^(x) u = 4x. . . .dv = e^(-x) du = 4. . . .v = - e^(-x) u * v - ? v * du 4x * -e^(-x) - ? 4 * -e^(-x) dx - 4x * e^(-x) + ? 4 e^(-x) dx - 4x * e^(-x) - 4 e^(-x) 2p * [ 4x * e^(x) - 4 * e^(x) - [ - 4x * e^(-x) - 4 e^(-x) ] ] 2p * [4x * e^(x) - 4 * e^(x) + 4x * e^(-x) + 4 e^(-x) ] -----> from 0 to 1 2p * [4 * (1 * e^(1) - 0 * e^(0) ) - 4 * ( e^(1) - e^(0) ) + 4 * (1 * e^(-1) - 0 * e^(-0)) + 4 * ( e^(-1) - e^(-0) )] 2p * [4 * ( e - 0 ) - 4 * ( e - 1 ) + 4 * ( 1/e - 0 ) + 4 * ( 1/e - 1 )] 2p * [4e - 4e + 4 + 4/e + 4/e - 4] 2p * [4/e + 4/e] 2p * [ 8/e ] 16p/e

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