# How do you find the volume of the solid generated by revolving the region bounded by the curves y = x² and y =1 rotated about the y=-2?

##### 1 Answer

See the explanation below.

#### Explanation:

Here is a picture of the region and a vertical slice.

The picture is set up to use washers (disks).

Thickness is

the radius of the larger washer is the greater

the radius of the smaller washer is the lesser

The representative slice has volume

We need to evaluate the integral

# = 104/15pi#

**Shells Method**

If we had taken a slice horizontally:

This is set up to use cylindrical shells of thickness

The volume of each shell is

In the region, the

The *radius* of the representative shell is

The *height* goes from the greater

We need to rewrite the boundary as functions of

The representative shell has volume

The solid has volume

# =4pi(26/15) = 104/15 pi#