Precalculus-What-is-the-volume-of-the-solid-obtained-by-rotating-the-region-bounded-by-x-y-2-2-and-y-x-about-the-line-y-1-by-the-method-of-cylindrical-shells-

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Question Number 134591 by bobhans last updated on 05/Mar/21

Precalculus What is the volume of the solid obtained by rotating the region bounded by x = (y−2)^2 and y = x about the line y = −1 by the method of cylindrical shells?

Precalculus

What is the volume of the solid obtained by rotating the region bounded by x = (y−2)^2 and y = x about the line y = −1 by the method of cylindrical shells?

Answered by EDWIN88 last updated on 05/Mar/21

Vol=2π∫_1 ^( 4) (y+1)(y−(y−2)^2 )dy V=2π∫_1 ^( 4) (y+1)(5y−y^2 −4)dy V=2π∫_1 ^( 4) (4y^2 +y−y^3 −4)dy V=2π [ ((4y^3 )/3)+(y^2 /2)−(y^4 /4)−4y ]_1 ^4 V=2π [((4(63))/3)+((15)/2)−((255)/4)−12 ] V=2π [ 72−((225)/4) ]= ((63π)/2)

Vol = 2 π \int_{1}^{4} (y + 1) (y - {(y - 2)}^{2}) dy

V = 2 π \int_{1}^{4} (y + 1) (5 y - y^{2} - 4) dy

V = 2 π \int_{1}^{4} ({4 y}^{2} + y - y^{3} - 4) dy

V = 2 π {[\frac{{4 y}^{3}}{3} + \frac{y^{2}}{2} - \frac{y^{4}}{4} - 4 y]}_{1}^{4}

V = 2 π [\frac{4 (63)}{3} + \frac{15}{2} - \frac{255}{4} - 12]

V = 2 π [72 - \frac{225}{4}] = \frac{63 π}{2}

Commented by bobhans last updated on 05/Mar/21

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Precalculus-What-is-the-volume-of-the-solid-obtained-by-rotating-the-region-bounded-by-x-y-2-2-and-y-x-about-the-line-y-1-by-the-method-of-cylindrical-shells-

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