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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. $x=3y^2$, $y=1$ and $x=0$ around the y-axis.

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  • $\begingroup$ It is $\frac {9 \pi} {5}$. $\endgroup$ – user98186 Nov 28 '15 at 23:32
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Draw a picture. The curve $x=3y^2$ is a rightward opening parabola with axis of symmetry the $x$-axis. We use cross-sections of the solid perpendicular to the $y$-axis.

The cross-sections are disks ("circles") of radius $x$. Thus the volume is $$\int_{y=0}^1 \pi x^2\,dy.$$ To evaluate, use the fact that $x=3y^2$.

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