# How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by y=x^3, x=1, y=0 revolved about the y-axis?

Dec 21, 2017

$V = 2 \pi r h \cdot \text{thickness} = 2 \pi \left(x\right) \left(y\right) \mathrm{dx} = 2 \pi {x}^{4} \mathrm{dx}$
$x$ varies from $0$ to $1$, so the solid has volume:
$V = {\int}_{0}^{1} 2 \pi {x}^{4} \mathrm{dx} = \frac{2 \pi}{5}$