# How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=x^2# and #y=2-x^2# and #x=0# about the line #x=1#?

##### 1 Answer

I would use shells (to avoid doing two integrals).

#### Explanation:

Here is a picture of the region. I have included the axis of rotation (

When we rotate the slice, we'll get a cylindrical shell.

The representative cylindrical shell will have volume:

#2 pixx"radius"xx"height"xx"thickness"# .

Here we have:

The height will be the greater

We also not that in the region,

Using shell, the volume of the solid is found by evaluating:

So we need

# = 2piint_0^1 (2-2x-2x^2+2x^3) dx#

# = 5/3 pi#