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