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?

1 Answer
Oct 25, 2015

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

enter image source here

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).