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Find the volume of the solid generated by rotating the following area around the y-axis.

The area will be bounded by $y=x^4$, $x=1$ and $y=0$.

What method is the easiest for this? I'm assuming shell method where the radius is $x$ and the height is $x^4$, but I'm not sure.

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  • $\begingroup$ In this case Shells is simpler than Slicing. You are well on your way to the answer. $\endgroup$ – André Nicolas Nov 3 '15 at 5:58
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Use the formula:

$\pi \int_0^1 dy R_1^2(y) - \pi \int_0^1 dy R_2^2(y)$

$R_1(y)$ is the $x=1$ bound.

$R_2(y)$ is the $y=x^4$ bound, which is more conveniently expressed as $x=y^{1/4}$.

This method integrates disks with radius $R(y)$.

You would then write:

$\pi \int_0^1 dy - \pi \int_0^1 dy (y^{1/4})^2$

$= \pi - \frac{2\pi}{3} $

$=\frac{\pi}{3}$

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