Question

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)^2` and `y=x^2-4x+7` rotated about the y axis?

Answer 1

Please see the explanation section, below.

Explanation:

`y=4(x-2)^2` and `y=x^2-4x+7`

The region bounded by the functions is shown below in blue.
We are using the shell method, so we take a representative slice parallel to the axis of revolution. In this case, that is the `y`-axis. So the thickness of the slice is `dx`.
The dashed line shows the radius of revolution for the representative.

We will be integrating with respect to `x`, so we need the minimum and maximum `x` values.

Solving: `4(x-2)^2 = x^2-4x+7`, we find the `x` values at the points of intersection to be `x=1,3`. So `x` varies from `1` to `3`.

The volume of a cylindrical shell is the surface area of the cylinder times the thickness:

`2pirh xx "thickness"`

In the picture, the radius is shown by the dashed line, it is `x`.

The height of the slice is the upper `y` value minus the lower: `(x^2-4x+7)-4(x-2)^2`

The volume of the representative is: `2pix[(x^2-4x+7)-4(x-2)^2]dx`

The volume of the resulting solid is:

`V=int_1^3 2pix[(x^2-4x+7)-4(x-2)^2]dx`

`= 2pi int_1^3 [-3x^3-4x^2+3x]dx`

`= 2pi[8] = 16pi`