The region R is bounded by the curves y=x² (x»0) , y=9x² (x»0) and the line y=1. Find the volume of the solid obtained when the region R is rotated about the y-axis?

1 Answer
Cosmic Defect
Dec 15, 2017

#V = \int_{y=0}^{y=1}\int_{x=\sqrt{y}/3}^{x=\sqrt{y}}(2\piy) dydx = (8\pi)/15.#

Explanation:

#V = \int_{y=0}^{y=1}\int_{x=\sqrt{y}/3}^{x=\sqrt{y}}(2\piy) dydx#

#V = 2\pi.\int_{y=0}^{y=1}y[\int_{x=\sqrt{y}/3}^{x=\sqrt{y}}dx]dy#

#V = 2\pi.\int_{y=0}^{y=1}y[\sqrt{y}-\sqrt{y}/3]dy=(4\pi)/3.\int_{0}^{1}y\sqrt{y}dy#

#V = (4\pi)/3. [2/5y^{5/2}]_0^1=(8\pi)/15.#

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