How To Find Volume Rotated About Y Axis

6. Applications of Integration

6.3 Volumes of Revolution: Cylindrical Shells

Learning Objectives

Calculate the volume of a solid of revolution by using the method of cylindrical shells.
Compare the different methods for calculating a volume of revolution.

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 bounded above by the graph of a function below by the $x\text{-axis,}$ and on the left and right by the lines and respectively, as shown in (Figure)(a). We then revolve this region around the -axis, as shown in (Figure)(b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of were revolved around the $x\text{-axis}$ or a line parallel to it.

As we have done many times before, partition the interval $\left[a,b\right]$ using a regular partition, $P=\left\{{x}_{0},{x}_{1}\text{,…},{x}_{n}\right\}$ and, for $i=1,2\text{,…},n,$ choose a point ${x}_{i}^{*}\in \left[{x}_{i-1},{x}_{i}\right].$ Then, construct a rectangle over the interval $\left[{x}_{i-1},{x}_{i}\right]$ of height $f({x}_{i}^{*})$ and width $\text{Δ}x.$ A representative rectangle is shown in (Figure)(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.

To calculate the volume of this shell, consider (Figure).

This figure is a graph in the first quadrant. The curve is increasing and labeled — Figure 3. Calculating 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 $\pi {x}_{i}^{2}-\pi {x}_{i-1}^{2}.$ The height of the cylinder is $f({x}_{i}^{*}).$ Then the volume of the shell is

$\begin{array}{cc}\hfill {V}_{\text{shell}}& =f({x}_{i}^{*})(\pi {x}_{i}^{2}-\pi {x}_{i-1}^{2})\hfill \\ & =\pi f({x}_{i}^{*})({x}_{i}^{2}-{x}_{i-1}^{2})\hfill \\ & =\pi f({x}_{i}^{*})({x}_{i}+{x}_{i-1})({x}_{i}-{x}_{i-1})\hfill \\ & =2\pi f({x}_{i}^{*})(\frac{{x}_{i}+{x}_{i-1}}{2})({x}_{i}-{x}_{i-1}).\hfill \end{array}$

Note that ${x}_{i}-{x}_{i-1}=\text{Δ}x,$ so we have

${V}_{\text{shell}}=2\pi f({x}_{i}^{*})(\frac{{x}_{i}+{x}_{i-1}}{2})\text{Δ}x.$

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

${V}_{\text{shell}}\approx 2\pi f({x}_{i}^{*}){x}_{i}^{*}\text{Δ}x.$

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)).

This figure has two images. The first is labeled — Figure 4. (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 $2\pi {x}_{i}^{*},$ and thickness $\text{Δ}x$ ((Figure)). 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

${V}_{\text{shell}}\approx f({x}_{i}^{*})(2\pi {x}_{i}^{*})\text{Δ}x,$

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

$V\approx \underset{i=1}{\overset{n}{\text{∑}}}(2\pi {x}_{i}^{*}f({x}_{i}^{*})\text{Δ}x).$

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

$V=\underset{n\to \infty }{\text{lim}}\underset{i=1}{\overset{n}{\text{∑}}}(2\pi {x}_{i}^{*}f({x}_{i}^{*})\text{Δ}x)={\int }_{a}^{b}(2\pi xf(x))dx.$

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

Now let's consider an example.

The Method of Cylindrical Shells 1

Define as the region bounded above by the graph of $f(x)=1\text{/}x$ and below by the $x\text{-axis}$ over the interval $\left[1,3\right].$ Find the volume of the solid of revolution formed by revolving around the $y\text{-axis}.$

Solution

First we must graph the region and the associated solid of revolution, as shown in the following figure.

Then the volume of the solid is given by

$\begin{array}{cc}\hfill V& ={\int }_{a}^{b}(2\pi xf(x))dx\hfill \\ & ={\int }_{1}^{3}(2\pi x(\frac{1}{x}))dx\hfill \\ & ={\int }_{1}^{3}2\pi dx={2\pi x|}_{1}^{3}=4\pi {\text{units}}^{3}\text{.}\hfill \end{array}$

The Method of Cylindrical Shells 2

Define R as the region bounded above by the graph of $f(x)=2x-{x}^{2}$ and below by the $x\text{-axis}$ over the interval $\left[0,2\right].$ Find the volume of the solid of revolution formed by revolving around the $y\text{-axis}.$

Solution

First graph the region and the associated solid of revolution, as shown in the following figure.

Then the volume of the solid is given by

$\begin{array}{cc}\hfill V& ={\int }_{a}^{b}(2\pi xf(x))dx\hfill \\ & ={\int }_{0}^{2}(2\pi x(2x-{x}^{2}))dx=2\pi {\int }_{0}^{2}(2{x}^{2}-{x}^{3})dx\hfill \\ & ={2\pi \left[\frac{2{x}^{3}}{3}-\frac{{x}^{4}}{4}\right]|}_{0}^{2}=\frac{8\pi }{3}{\text{units}}^{3}\text{.}\hfill \end{array}$

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

The Method of Cylindrical Shells for a Solid Revolved around the -axis

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

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 $y\text{-axis},$ we have a different outer and inner radius. Suppose, for example, that we rotate the region around the line $x=\text{−}k,$ where is some positive constant. Then, the outer radius of the shell is ${x}_{i}+k$ and the inner radius of the shell is ${x}_{i-1}+k.$ Substituting these terms into the expression for volume, we see that when a plane region is rotated around the line $x=\text{−}k,$ the volume of a shell is given by

$\begin{array}{cc}\hfill {V}_{\text{shell}}& =2\pi f({x}_{i}^{*})(\frac{({x}_{i}+k)+({x}_{i-1}+k)}{2})(({x}_{i}+k)-({x}_{i-1}+k))\hfill \\ & =2\pi f({x}_{i}^{*})((\frac{{x}_{i}+{x}_{i-2}}{2})+k)\text{Δ}x.\hfill \end{array}$

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

${V}_{\text{shell}}\approx 2\pi ({x}_{i}^{*}+k)f({x}_{i}^{*})\text{Δ}x.$

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

$V={\int }_{a}^{b}(2\pi (x+k)f(x))dx.$

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 $x\text{-term}$ in the integral must be replaced with an expression representing the radius of a shell. To see how this works, consider the following example.

A Region of Revolution Revolved around a Line

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.

A Region of Revolution 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) describes the different approaches for solids of revolution around the $x\text{-axis}.$ It's up to you to develop the analogous table for solids of revolution around the $y\text{-axis}.$

This figure is a table comparing the different methods for finding volumes of solids of revolution. The columns in the table are labeled — Figure 10.

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

Selecting the Best Method

Solution

First, sketch the region and the solid of revolution as shown.
Looking at the region, if we want to integrate with respect to we would have to break the integral into two pieces, because we have different functions bounding the region over $\left[0,1\right]$ and $\left[1,2\right].$ In this case, using the disk method, we would have

$V={\int }_{0}^{1}(\pi {x}^{2})dx+{\int }_{1}^{2}(\pi {(2-x)}^{2})dx.$

If we used the shell method instead, we would use functions of to represent the curves, producing

$\begin{array}{cc}\hfill V& ={\int }_{0}^{1}(2\pi y\left[(2-y)-y\right])dy\hfill \\ & ={\int }_{0}^{1}(2\pi y\left[2-2y\right])dy.\hfill \end{array}$

Neither of these integrals is particularly onerous, but since the shell method requires only one integral, and the integrand requires less simplification, we should probably go with the shell method in this case.
First, sketch the region and the solid of revolution as shown.
Looking at the region, it would be problematic to define a horizontal rectangle; the region is bounded on the left and right by the same function. Therefore, we can dismiss the method of shells. The solid has no cavity in the middle, so we can use the method of disks. Then

$V={\int }_{0}^{4}\pi {(4x-{x}^{2})}^{2}dx.$

Select the best method to find the volume of a solid of revolution generated by revolving the given region around the $x\text{-axis},$ and set up the integral to find the volume (do not evaluate the integral): the region bounded by the graphs of $y=2-{x}^{2}$ and $y={x}^{2}.$

Solution

Use the method of washers; $V={\int }_{-1}^{1}\pi \left[{(2-{x}^{2})}^{2}-{({x}^{2})}^{2}\right]dx$

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
$V={\int }_{a}^{b}(2\pi xf(x))dx$

For the following exercise, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.

2. [T] Under the curve of $y=3x,x=0,\text{ and }x=3$ rotated around the $y\text{-axis}.$

Solution

This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.
$54\pi$ units³

3. [T] Over the curve of $y=3x,x=0,\text{ and }y=3$ rotated around the $x\text{-axis}.$

4. [T] Under the curve of $y=3x,x=0,\text{ and }x=3$ rotated around the $x\text{-axis}.$

Solution

This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.
$81\pi$ units³

5. [T] Under the curve of $y=2{x}^{3},x=0,\text{ and }x=2$ rotated around the $y\text{-axis}.$

6. [T] Under the curve of $y=2{x}^{3},x=0,\text{ and }x=2$ rotated around the $x\text{-axis}.$

Solution

This figure is a graph in the first quadrant. It is the increasing curve y=2x^3. Under the curve and above the x-axis there is a shaded region. The region is bounded to the right at x=2.
$\frac{512\pi }{7}$ units³

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the $x\text{-axis}$ and are rotated around the $y\text{-axis}.$

7. $y=1-{x}^{2},x=0,\text{ and }x=1$

8. $y=5{x}^{3},x=0,\text{ and }x=1$

Solution

$2\pi$ units³

9. $y=\frac{1}{x},x=1,\text{ and }x=100$

10. $y=\sqrt{1-{x}^{2}},x=0,\text{ and }x=1$

Solution

$\frac{2\pi }{3}$ units³

11. $y=\frac{1}{1+{x}^{2}},x=0,\text{ and }x=3$

12. $y= \sin {x}^{2},x=0,\text{ and }x=\sqrt{\pi }$

Solution

$2\pi$ units³

13. $y=\frac{1}{\sqrt{1-{x}^{2}}},x=0,\text{ and }x=\frac{1}{2}$

14. $y=\sqrt{x},x=0,\text{ and }x=1$

Solution

$\frac{4\pi }{5}$ units³

15. $y={(1+{x}^{2})}^{3},x=0,\text{ and }x=1$

16. $y=5{x}^{3}-2{x}^{4},x=0,\text{ and }x=2$

Solution

$\frac{64\pi }{3}$ units³

For the following exercises, use shells to find the volume generated by rotating the regions between the given curve and around the $x\text{-axis}.$

17. $y=\sqrt{1-{x}^{2}},x=0,\text{ and }x=1$

18. $y={x}^{2},x=0,\text{ and }x=2$

Solution

$\frac{32\pi }{5}$ units³

19. $y={e}^{x},x=0,\text{ and }x=1$

20. $y=\text{ln}(x),x=1,\text{ and }x=e$

Solution

$\pi (e-2)$ units³

21. $x=\frac{1}{1+{y}^{2}},y=1,\text{ and }y=4$

22. $x=\frac{1+{y}^{2}}{y},y=0,\text{ and }y=2$

Solution

$\frac{28\pi }{3}$ units³

23. $x= \cos y,y=0,\text{ and }y=\pi$

24. $x={y}^{3}-4{y}^{2},x=-1,\text{ and }x=2$

Solution

$\frac{-84\pi }{5}$ units³

25. $x=y{e}^{y}\text{,}x=-1,\text{ and }x=2$

26. $x= \cos y{e}^{y},x=0,\text{ and }x=\pi$

Solution

$\text{−}{e}^{\pi }{\pi }^{2}$ units³

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis.

27. $y=3-x,y=0,x=0,\text{ and }x=2$ rotated around the $y\text{-axis}.$

28. $y={x}^{3},y=0,\text{ and }y=8$ rotated around the $y\text{-axis}.$

Solution

$\frac{64\pi }{5}$ units³

29. $y={x}^{2},y=x,$ rotated around the $y\text{-axis}.$

30. $y=\sqrt{x},x=0,\text{ and }x=1$ rotated around the line

Solution

$\frac{28\pi }{15}$ units³

31. $y=\frac{1}{4-x},x=1,\text{ and }x=2$ rotated around the line

32. $y=\sqrt{x}\text{ and }y={x}^{2}$ rotated around the $y\text{-axis}.$

Solution

$\frac{3\pi }{10}$ units³

33. $y=\sqrt{x}\text{ and }y={x}^{2}$ rotated around the line

34. $x={y}^{3},y=\frac{1}{x},x=1,\text{ and }y=2$ rotated around the $x\text{-axis}.$

Solution

$\frac{52\pi }{5}$ units³

35. $x={y}^{2}\text{ and }y=x$ rotated around the line

For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.

38. [T] $y= \cos (\pi x),y= \sin (\pi x),x=\frac{1}{4},\text{ and }x=\frac{5}{4}$ rotated around the $y\text{-axis}.$

Solution

This figure is a graph. On the graph are two curves, y=cos(pi times x) and y=sin(pi times x). They are periodic curves resembling waves. The curves intersect in the first quadrant and also the fourth quadrant. The region between the two points of intersection is shaded.
$3\sqrt{2}$ units³

39. [T] $y={x}^{2}-2x,x=2,\text{ and }x=4$ rotated around the $y\text{-axis}.$

40. [T] $y={x}^{2}-2x,x=2,\text{ and }x=4$ rotated around the $x\text{-axis}.$

Solution

This figure is a graph in the first quadrant. It is the parabola y=x^2-2x. . Under the curve and above the x-axis there is a shaded region. The region begins at x=2 and is bounded to the right at x=4.
$\frac{496\pi }{15}$ units³

41. [T] $y=3{x}^{3}-2,y=x,\text{ and }x=2$ rotated around the $x\text{-axis}.$

42. [T] $y=3{x}^{3}-2,y=x,\text{ and }x=2$ rotated around the $y\text{-axis}.$

Solution

This figure is a graph in the first quadrant. There are two curves on the graph. The first curve is y=3x^2-2 and the second curve is y=x. Between the curves there is a shaded region. The region begins at x=1 and is bounded to the right at x=2.
$\frac{398\pi }{15}$ units³

44. [T] $x={y}^{2},x={y}^{2}-2y+1,\text{ and }x=2$ rotated around the $y\text{-axis}.$

Solution

This figure is a graph. There are two curves on the graph. The first curve is x=y^2-2y+1 and is a parabola opening to the right. The second curve is x=y^2 and is a parabola opening to the right. Between the curves there is a shaded region. The shaded region is bounded to the right at x=2.
15.9074 units³

For the following exercises, use the method of shells to approximate the volumes of some common objects, which are pictured in accompanying figures.

45. Use the method of shells to find the volume of a sphere of radius

This figure has two images. The first is a circle with radius r. The second is a basketball.

46.Use the method of shells to find the volume of a cone with radius and height

This figure has two images. The first is an upside-down cone with radius r and height h. The second is an ice cream cone.

Solution

$\frac{1}{3}\pi {r}^{2}h$ units³

47.Use the method of shells to find the volume of an ellipse $({x}^{2}\text{/}{a}^{2})+({y}^{2}\text{/}{b}^{2})=1$ rotated around the $x\text{-axis}.$

This figure has two images. The first is an ellipse with a the horizontal distance from the center to the edge and b the vertical distance from the center to the top edge. The second is a watermelon.

48.Use the method of shells to find the volume of a cylinder with radius and height

This figure has two images. The first is a cylinder with radius r and height h. The second is a cylindrical candle.

Solution

$\pi {r}^{2}h$ units³

49.Use the method of shells to find the volume of the donut created when the circle ${x}^{2}+{y}^{2}=4$ is rotated around the line

This figure has two images. The first has two ellipses, one inside of the other. The radius of the path between them is 2 units. The second is a doughnut.

Glossary

method of cylindrical shells: a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable