Question

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?

Answer 1

`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.`