## Answers

See the explanation below.

#### Explanation:

I interpreted the question to include the region both above and below the

I've set it up to use washers, the representative slice is taken perpendicular to the axis of rotation and is shown in black. The dashed red lines above and below the slice show the greater and lesser radii of the washer. Call the

The thickness of the slice is

The volume of the representative washer is

We see that

The volume of the resulting solid is

# = pi int_-1^1 (11-12y^4+y^8)dy#

We can simplify the arithematic a bit be using symmetry of the region and evaluating from

# = 2pi(392/45) = (784pi)/45 ~~ 54.734#