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 `x=y-y^2` and the y axis rotated about the y axis?

Answer 1

See the explanation section, below.

Explanation:

Here is a picture of the region. Because we have been asked to ue shells, a representative slice has been taken parallel to the axis of revolution. This make the thickness of our slice `dx`

The volume of the representative shell is:

`2 pixx "radius"xx"height"xx"thickness"`

As mentioned, `"thickness" = dx`
and we can see that `"radius"=x`

The `"height"` will be the greater `y` value minus the lesser `y` value. We are working in terms of `x`, so we need to write these two `y` values as functions of `x`.

We need to solve `x=y-y^2` for `y` (in terms of `x`).

There are a couple of ways of doing this. (1) complete the square or (2) use the quadratic formula to solve `y^2-y+x=0` for `y`.
(There are other ways as well. For example, you can use the vertex formula to write the equation in standard form for a sideways opening parabola.)

I'll complete the square.

`y^2-y = -x`

`y^2-y +1/4 = 1/4-x`

`(y-1/2)^2 = (1-4x)/4`

`y-1/2 = +-sqrt((1-4x)/4)`

`y = 1/2+-sqrt(1-4x)/2 = (1+-sqrt(1-4x))/2`

(Yes, this is the same as the answer you'll get by using the quadratic formula.)

The greater `y` (the one on top) is
`y_"top" = (1+sqrt(1-4x))/2`

and the lesser `y` (the one on the bottom) is
`y_"bottom" = (1-sqrt(1-4x))/2`

So, finally, we can write the volume of the representative cylindrical shell:

`2 pixx "radius"xx"height"xx"thickness" = 2pix((1+sqrt(1-4x))/2-(1-sqrt(1-4x))/2) dx`

Don't Panic. We can simplify this to get

`2pixsqrt(1-4x)dx`

We still haven't found the bounds on `x`.

In the region we have `0 <= x`, and, examining the solution for `y`, we see that if `x > 1/4` we'll get imaginary values for `y`. So, we get `x` varies from `0` to `1/4`.

The volume of the solid of interest is

`V = int_0^(1/4) 2 pi x sqrt(1-4x) dx`

Which can be evaluated by parts or by the substitution `u = 1-4x` so that `du = -4 dx` and `x = 1/4(1-u)`.