6.3 Volumes of Revolution: Cylindrical Shells

In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method; however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. In the last part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to help you determine which method to use in a given situation.

The Method of Cylindrical Shells

Again, we are working with a solid of revolution. As before, we define a region $R,$ bounded above by the graph of a function $y = f (x),$ below by the $x -axis,$ and on the left and right by the lines $x = a$ and $x = b,$ respectively, as shown in Figure 6.25(a). We then revolve this region around the y-axis, as shown in Figure 6.25(b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of $x$ were revolved around the $x -axis$ or a line parallel to it.

This figure has two graphs. The first graph is labeled “a” and is an increasing curve in the first quadrant. The curve is labeled “y=f(x)”. The curve starts on the y-axis at y=a. Under the curve, above the x-axis is a shaded region labeled “R”. The shaded region is bounded on the right by the line x=b. The second graph is a three dimensional solid. It has been created by rotating the shaded region from “a” around the y-axis. — Figure 6.25(a) A region bounded by the graph of a function of $x .$ (b) The solid of revolution formed when the region is revolved around the $y -axis .$

As we have done many times before, partition the interval $[a, b]$ using a regular partition, $P = {x_{0}, x_{1}, \dots, x_{n}}$ and, for $i = 1, 2, \dots, n,$ choose a point Then, construct a rectangle over the interval of height and width A representative rectangle is shown in Figure 6.26(a). When that rectangle is revolved around the -axis, instead of a disk or a washer, we get a cylindrical shell, as shown in the following figure.

This figure has two images. The first is a cylindrical shell, hollow in the middle. It has a vertical axis in the center. There is also a curve that meets the top of the cylinder. The second image is a set of concentric cylinders, one inside of the other forming a nesting of cylinders. — Figure 6.26(a) A representative rectangle. (b) When this rectangle is revolved around the $y -axis,$ the result is a cylindrical shell. (c) When we put all the shells together, we get an approximation of the original solid.

To calculate the volume of this shell, consider Figure 6.27.

This figure is a graph in the first quadrant. The curve is increasing and labeled “y=f(x)”. The curve starts on the y-axis at f(x*). Below the curve is a shaded rectangle. The rectangle starts on the x-axis. The width of the rectangle is delta x. The two sides of the rectangle are labeled “xsub(i-1)” and “xsubi”. — Figure 6.27Calculating the volume of the shell.

The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. The cross-sections are annuli (ring-shaped regions—essentially, circles with a hole in the center), with outer radius $x_{i}$ and inner radius $x_{i - 1} .$ Thus, the cross-sectional area is $π x_{i}^{2} - π x_{i - 1}^{2} .$ The height of the cylinder is Then the volume of the shell is

V_{shell} = f (x_{i}^{*}) (π x_{i}^{2} - π x_{i - 1}^{2}) = π f (x_{i}^{*}) (x_{i}^{2} - x_{i - 1}^{2}) = π f (x_{i}^{*}) (x_{i} + x_{i - 1}) (x_{i} - x_{i - 1}) = 2 π f (x_{i}^{*}) (\frac{x _{i} + x _{i - 1}}{2}) (x_{i} - x_{i - 1}) .

Note that $x_{i} - x_{i - 1} = Δ x,$ so we have

V_{shell} = 2 π f (x_{i}^{*}) (\frac{x _{i} + x _{i - 1}}{2}) Δ x .

Furthermore, $\frac{x _{i} + x _{i - 1}}{2}$ is both the midpoint of the interval $[x_{i - 1}, x_{i}]$ and the average radius of the shell, and we can approximate this by We then have

V_{shell} \approx 2 π f (x_{i}^{*}) x_{i}^{*} Δ x .

Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate (Figure 6.28).

This figure has two images. The first is labeled “a” and is of a hollow cylinder around the y-axis. On the front of this cylinder is a vertical line labeled “cut line”. The height of the cylinder is “y=f(x)”. The second figure is labeled “b” and is a shaded rectangular block. The height of the rectangle is “f(x*), the width of the rectangle is “2pix*”, and the thickness of the rectangle is “delta x”. — Figure 6.28(a) Make a vertical cut in a representative shell. (b) Open the shell up to form a flat plate.

In reality, the outer radius of the shell is greater than the inner radius, and hence the back edge of the plate would be slightly longer than the front edge of the plate. However, we can approximate the flattened shell by a flat plate of height $f (x_{i}^{*}),$ width $2 π x_{i}^{*},$ and thickness $Δ x$ (Figure 6.28). The volume of the shell, then, is approximately the volume of the flat plate. Multiplying the height, width, and depth of the plate, we get

V_{shell} \approx f (x_{i}^{*}) (2 π x_{i}^{*}) Δ x,

which is the same formula we had before.

To calculate the volume of the entire solid, we then add the volumes of all the shells and obtain

V \approx i = 1 \sum n (2 π x_{i}^{*} f (x_{i}^{*}) Δ x) .

Here we have another Riemann sum, this time for the function $2 π x f (x) .$ Taking the limit as $n \to \infty$ gives us

V = n \to \infty lim i = 1 \sum n (2 π x_{i}^{*} f (x_{i}^{*}) Δ x) = \int_{a}^{b} (2 π x f (x)) d x .

This leads to the following rule for the method of cylindrical shells.

Now let’s consider an example.

As with the disk method and the washer method, we can use the method of cylindrical shells with solids of revolution, revolved around the $x -axis,$ when we want to integrate with respect to $y .$ The analogous rule for this type of solid is given here.

For the next example, we look at a solid of revolution for which the graph of a function is revolved around a line other than one of the two coordinate axes. To set this up, we need to revisit the development of the method of cylindrical shells. Recall that we found the volume of one of the shells to be given by

V_{shell} = f (x_{i}^{*}) (π x_{i}^{2} - π x_{i - 1}^{2}) = π f (x_{i}^{*}) (x_{i}^{2} - x_{i - 1}^{2}) = π f (x_{i}^{*}) (x_{i} + x_{i - 1}) (x_{i} - x_{i - 1}) = 2 π f (x_{i}^{*}) (\frac{x _{i} + x _{i - 1}}{2}) (x_{i} - x_{i - 1}) .

This was based on a shell with an outer radius of $x_{i}$ and an inner radius of $x_{i - 1} .$ If, however, we rotate the region around a line other than the $y -axis,$ we have a different outer and inner radius. Suppose, for example, that we rotate the region around the line $x = - k,$ where $k$ is some positive constant. Then, the outer radius of the shell is and the inner radius of the shell is Substituting these terms into the expression for volume, we see that when a plane region is rotated around the line the volume of a shell is given by

V_{shell} = 2 π f (x_{i}^{*}) (\frac{( x _{i} + k ) + ( x _{i - 1} + k )}{2}) ((x_{i} + k) - (x_{i - 1} + k)) = 2 π f (x_{i}^{*}) ((\frac{x _{i} + x _{i - 1}}{2}) + k) Δ x .

As before, we notice that $\frac{x _{i} + x _{i - 1}}{2}$ is the midpoint of the interval $[x_{i - 1}, x_{i}]$ and can be approximated by Then, the approximate volume of the shell is

V_{shell} \approx 2 π (x_{i}^{*} + k) f (x_{i}^{*}) Δ x .

The remainder of the development proceeds as before, and we see that

V = \int_{a}^{b} (2 π (x + k) f (x)) d x .

We could also rotate the region around other horizontal or vertical lines, such as a vertical line in the right half plane. In each case, the volume formula must be adjusted accordingly. Specifically, the $x -term$ in the integral must be replaced with an expression representing the radius of a shell. To see how this works, consider the following example.

For our final example in this section, let’s look at the volume of a solid of revolution for which the region of revolution is bounded by the graphs of two functions.

Which Method Should We Use?

We have studied several methods for finding the volume of a solid of revolution, but how do we know which method to use? It often comes down to a choice of which integral is easiest to evaluate. Figure 6.34 describes the different approaches for solids of revolution around the $x -axis .$ It’s up to you to develop the analogous table for solids of revolution around the $y -axis .$

This figure is a table comparing the different methods for finding volumes of solids of revolution. The columns in the table are labeled “comparison”, “disk method”, “washer method”, and “shell method”. The rows are labeled “volume formula”, “solid”, “interval to partition”, “rectangles”, “typical region”, and “rectangle”. In the disk method column, the formula is given as the definite integral from a to b of pi times [f(x)]^2. The solid has no cavity in the center, the partition is [a,b], rectangles are vertical, and the typical region is a shaded region above the x-axis and below the curve of f(x). In the washer method column, the formula is given as the definite integral from a to b of pi times [f(x)]^2-[g(x)]^2. The solid has a cavity in the center, the partition is [a,b], rectangles are vertical, and the typical region is a shaded region above the curve of g(x) and below the curve of f(x). In the shell method column, the formula is given as the definite integral from c to d of 2pi times yg(y). The solid is with or without a cavity in the center, the partition is [c,d] rectangles are horizontal, and the typical region is a shaded region above the x-axis and below the curve of g(y). — Figure 6.34

Let’s take a look at a couple of additional problems and decide on the best approach to take for solving them.

Key Concepts

The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. This method is sometimes preferable to either the method of disks or the method of washers because we integrate with respect to the other variable. In some cases, one integral is substantially more complicated than the other.
The geometry of the functions and the difficulty of the integration are the main factors in deciding which integration method to use.

Key Equations

Method of Cylindrical Shells

V = \int_{a}^{b} (2 π x f (x)) d x