Precalculus-How-do-I-find-the-volume-of-a-solid-obtained-by-rotating-the-region-bounded-by-x-y-3-2-and-x-4-about-y-1-

Question Number 134587 by bobhans last updated on 05/Mar/21

Precalculus

How do I find the volume of a solid obtained by rotating the region bounded by x=(y−3)^2 and x=4 about y=1?

Commented by EDWIN88 last updated on 05/Mar/21

Commented by EDWIN88 last updated on 05/Mar/21

Vol = 2 π \int_{1}^{5} (y - 1) (4 - {(y - 3)}^{2}) dy

Vol = 2 π \int_{1}^{5} (y - 1) (- y^{2} + 6 y - 5) dy

Vol = 2 π \int_{1}^{5} (- y^{3} + {7 y}^{2} - 11 y + 5) dy

Vol = 2 π {[- \frac{y^{4}}{4} + \frac{{7 y}^{3}}{3} - \frac{{11 y}^{2}}{2} + 5 y]}_{1}^{5}

Vol = 2 π [- (\frac{624}{4}) + \frac{7 (124)}{3} - \frac{11 (24)}{2} + 20]

Vol = 2 π (- 156 + \frac{868}{3} - 132 + 20)

Vol = 2 π (\frac{868}{3} - \frac{804}{3}) = \frac{128 π}{3}

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