Question

How do you find the volume of the region enclosed by the curves `y=x`, `y=-x`, and `x=1` rotated about `y=1`?

Answer 1

The volume is `2pi`

Explanation:

Using the method of washers

Let the outer radius be `1-(-x)=1+x`

Let the inner radius be `1-x`

The integral for the volume is

`piint_0^1(1+x)^2-(1-x)^2dx`

`piint_0^1(1+2x+x^2)-(1-2x+x^2)dx`

`piint_0^1(1+2x+x^2-1+2x-x^2)dx`

`piint_0^1(4x)dx`

Integrating we get

`2pix^2`

Evaluating from `0` to `1`

`2pi(1)^2-0=2pi`