y = 1/?(1+ x^2 ), x = 0, and x = 1 is rotated about the x-axis. Solution using t
ID: 3140029 • Letter: Y
Question
y = 1/?(1+ x^2 ), x = 0, and x = 1 is rotated about the x-axis.Explanation / Answer
using the Washer Method: about x-axis y = 3x^2 y = 5x + 2 limits: 3x^2 = 5x + 2 3x^2 - 5x - 2 = 0 (3x + 1)(x - 2) = 0 ----> x = -1/3 & 2 A (x) = p ( outer radius )^2 - p ( inner radius )^2 A (x) = p ( 5x + 2 - 0 )^2 - p ( 3x^2 - 0 )^2 A (x) = p ( 5x + 2 )^2 - p ( 3x^2 )^2 A (x) = p ( 25x^2 + 2*5x*2 + 4 - 9x^4 ) A (x) = p ( 25x^2 + 20x + 4 - 9x^4 ) 2 ? p ( 25x^2 + 20x + 4 - 9x^4 ) dx -1/3 . . . . . .. . . . . . . . .. . . . .. . . .. . . ... . . . . 2 p * [ (25/3) * x^3 + 10x^2 + 4x - (9/5) * x^5 ] . . . . . .. . . . . . . . .. . . . .. . . .. . . . . . . . -1/3 p * [ (25/3) * (2^3 - (-1/3)^3) + 10 * (2^2 - (-1/3)^2) + 4 * (2 - (-1/3)) - (9/5) * (2^5 - (-1/3)^5) ] p * [ (25/3) * ( 8 - (-1/27) ) + 10 * ( 4 - (1/9) ) + 4 * ( 2 + (1/3) ) - (9/5) * ( 32 - (-1/243) ) ] p * [ (25/3) * ( 8 + (1/27) ) + 10 * (35/9) + 4 * (7/3) - (9/5) * ( 32 + (1/243) ) ] p * [ (25/3) * (217/27) + (350/9) + (28/3) - (9/5) * (7777/243) ] p * [ (5425/81) + (350/9) + (28/3) - (7777/135) ] (23324p/405) Using shell method: y = 3x^2. . . . . x = +/- v(y/3) y = 5x + 2. . . . x = (y - 2)/5 height ===> v(y/3) - (y - 2)/5 radius ====> y limits: when x = -1/3 ----> y = 1/3 when x = 2 ----> y = 12 12. . . . . . . . . . . .. . . . . .12 ? 2p * y * ( v(y/3) - 0 ) dy - ? 2p * y * ( (y - 2)/5 - 0 ) dy = 856p/15 0. . . . . . . . . . . .. . . . . . .2 1/3. . . . . . . . . . .. . . . . . 2 ? 2p * y * ( 0 - -v(y/3) ) dy + ? 2p * y * ( 0 - (y - 2)/5 ) dy = 212p/405 0. . . . . . . . . . . .. . . . . . . 1/3 12 212p/405 + 856p/15 = 23324p/405
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