[Math] Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

calculus

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.

Best Answer

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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