Volume by Shells

Volumes of Revolution by Shells

We consider revolving the region bounded by the x axis and the graph of y = 4x - x² about the y axis. The picture below (which is animated if your Web browser supports animation) illustrates the object whose volume we want to find. Animated image of a region being revolved about the y axis to form a solid.

Animated image of a region being revolved about the y axis to form a solid.

As usual, our approach is to divide and conquer. We focus on a small rectangular slice of the area, whose width is dx a "small amount of x." In addition, the presence of dx tells us that x is the variable of integration. So we will have to express everything in terms of the variable x before we can carry out the integration.

This slice must be a general slice, which will cover the entire region of interest as x varies between some upper and lower limits. So the height of this slice must vary as well. The height is measured vertically, so we take the y value at the top of the slice (y = 4x - x²) and subtract the y value at the bottom of the slice (y = 0).(Q: Do we measure the height at the left or right side of the slice? A: It doesn't matter.)

When we rotate the region about the y axis, this small slice of area (which we called dA before) will become a small piece of volume, called a cylindrical shell. The radius of this cylindrical shell is always the distance from the axis of revolution. In this case, that radius is measured horizontally, so it is the x value at the slice minus the x value at the axis of revolution. Since we are revolving about the y axis, the x value there is zero. (Q: Do we measure the radius to the left or right side of the slice? A: It doesn't matter.)

Press the button below to view the animation.

The key to finding the volume of a cylindrical shell is to see it as a rectangular shape, whose volume is simply length times height times thickness. The length of the shell is 2πr, where r is the radius. The thickness will always be either dx or dy, and the height will be either a difference of y values, or a difference of x values, depending on how the shell is oriented. Then the small piece of volume represented by the shell is always

dV = (2πr)(h)(thickness).

Depending on the particular axis of revolution, and the way the curve is oriented, there are about eight different ways this volume expression can turn out. But just remember it in this general form, and work out exactly what the height, thickness and radius are in each individual case.

In the case illustrated above, we have

r = x
h = 4x - x²
thickness = dx
volume dV = 2πx(4x - x²)dx

Integration is the process of adding up all the small pieces dV of volume. The variable of integration is x, so we need the smallest and largest values of x involved in the region. These clearly occur where the graph crosses the x axis, at 0 and 4.

Finally, integration is the process of adding up all the small pieces dV of volume to get the total volume of the solid:

Displayed equation computing the integral from 0 to 4 of 2 pi x times 4x minus x squared, with respect to x. The value of the integral is 128 thirds pi.

As always, we should make a quick check that this is reasonable. This volume is about 128, and the object in question would fit inside a cylinder of radius 4 and height 4. That cylinder has volume V = πr²h = 64 π, or about 192. So the solid of revolution fills two-thirds of the cylinder, which is reasonable.

What we can ignore When setting up the integration, we think of dx as a "small amount of x." You should recall, however, that the integral involves a limit as dx goes to zero. Our sketches, of course, always show a particular non-zero dx. When setting up the integral, we can ignore anything which will go to zero when dx goes to zero. It doesn't matter whether we call the radius x or x + dx; we choose x because it produces a simpler expression. Similarly, the height could be f(x) or f(x + dx); we choose the simpler one. [ Close ]

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