# Basis of the Shell Method

In order to find the volume of that curve rotated around the y-axis, you need the radius $(x)$, the height $(f(x))$, and the thickness is the change in x, or $\Delta$x.

So, the volume of the whole thing is:

# Example Problem

Radius = $1 - x$

This is because the radius is how far away the $x$ is from $x = 1$, so that's $1 - x$.

Height = $f(x)$, or $1 - x^2$

This is pretty straightforward, at any given shell, the height is $f(x)$. So the v = $$2\pi\int_{-1}^{1}(1-x)(1-x^2)\, \mathrm{d}x$$ You can expand, then you get $$\boxed{v = \dfrac{8\pi}{3}} = 8.3776$$

David Witten