# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region y=4-x^2 and y=0 rotated about the y-axis?

Oct 14, 2015

See the explanation, below.

#### Explanation:

Here is the region with a thin slice taken parallel to the axis of rotation. (To set up for cylindrical shells.)

The volume of a representative shell is $2 \pi r h \cdot \text{thickness}$

In this case, we have radius $r = x$, height $h = 4 - {x}^{2}$ and $\text{thickness} = \mathrm{dx}$. $x$ varies from $0$ to $2$, so the volume of the solid is:

${\int}_{0}^{2} \pi x \left(4 - {x}^{2}\right) \mathrm{dx} = \pi {\int}_{0}^{2} \left(4 x - {x}^{3}\right) \mathrm{dx} = 4 \pi$