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{\bf Question}

Sketch the region enclosed by the given curves and find the volume
of the solid generated when it is revolved about the $x$-axis:

 $x=2y$, $y=1$, $y=3$, $x=0$.

{\bf Answer}

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Best to use shell method:

$$ \int_{y=1}^{y=3} (2\pi y)(2y) \,dy = 4\pi \int_{y=1}^{y=3}y^2
\,dy = 4\pi \left[\frac{y^3}{3} \right]_1^3 = 4\pi
\left(9-\frac{1}{3} \right)=\frac{104\pi}{3}$$


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