Estimate the Volume if the Region Is Rotated About the Y Axis Again Use the Midpoint Rule With N 4

vi. 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 computing a volume of revolution.

In this department, nosotros examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can utilise this method on the same kinds of solids as the disk method or the washer method; even so, with the disk and washer methods, we integrate forth the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, nosotros integrate along the coordinate centrality perpendicular to the centrality of revolution. The power to cull which variable of integration we want to utilize can be a pregnant 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 final part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to assist you make up one's mind which method to use in a given situation.

The Method of Cylindrical Shells

Again, we are working with a solid of revolution. Equally before, we define a region $R,$ bounded above by the graph of a function $y=f(x),$ below by the $x\text{-axis,}$ and on the left and right by the lines $x=a$ and $x=b,$ respectively, equally shown in (Figure)(a). Nosotros then revolve this region around the $y$ -axis, as shown in (Figure)(b). Notation that this is dissimilar from what we accept washed before. Previously, regions defined in terms of functions of $x$ were revolved effectually the $x\text{-axis}$ or a line parallel to it.

Every bit we have done many times before, sectionalisation the interval $\left[a,b\right]$ using a regular segmentation, $P=\left\{{x}_{0},{x}_{1}\text{,…},{x}_{n}\right\}$ and, for $i=1,2\text{,…},n,$ choose a point ${x}_{i}^{*}\in \left[{x}_{i-1},{x}_{i}\right].$ And then, construct a rectangle over the interval $\left[{x}_{i-1},{x}_{i}\right]$ of top $f({x}_{i}^{*})$ and width $\text{Δ}x.$ A representative rectangle is shown in (Figure)(a). When that rectangle is revolved around the $y$ -centrality, instead of a disk or a washer, we get a cylindrical shell, as shown in the post-obit figure.

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

This figure is a graph in the first quadrant. The curve is increasing and labeled — Figure 3. Computing the volume of the shell.

The shell is a cylinder, so its volume is the cantankerous-sectional area multiplied by the pinnacle of the cylinder. The cross-sections are annuli (band-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 surface 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

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

Notation that ${x}_{i}-{x}_{i-1}=\text{Δ}x,$ so nosotros 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 fashion to think of this is to think of making a vertical cut in the shell and so opening it up to class a flat plate ((Figure)).

This figure has two images. The first is labeled — Effigy 4. (a) Make a vertical cutting in a representative vanquish. (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 end border of the plate. However, we can approximate the flattened crush by a flat plate of elevation $f({x}_{i}^{*}),$ width $2\pi {x}_{i}^{*},$ and thickness $\text{Δ}x$ ((Figure)). The volume of the beat, then, is approximately the volume of the apartment 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 book of the entire solid, nosotros 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 nosotros 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 post-obit rule for the method of cylindrical shells.

Now let's consider an example.

The Method of Cylindrical Shells 1

Define $R$ every bit the region divisional to a higher place by the graph of $f(x)=1\text{/}x$ and below by the $x\text{-axis}$ over the interval $\left[1,3\right].$ Observe the book of the solid of revolution formed by revolving $R$ around the $y\text{-axis}.$

Solution

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

And so 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 two

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

Solution

Kickoff graph the region $R$ and the associated solid of revolution, every bit shown in the post-obit effigy.

And so 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 effectually the $x\text{-axis},$ when nosotros desire to integrate with respect to $y.$ The coordinating dominion for this type of solid is given hither.

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

For the next example, we await at a solid of revolution for which the graph of a function is revolved around a line other than i of the two coordinate axes. To set this upwards, nosotros need to revisit the evolution of the method of cylindrical shells. Recall that we found the book of one of the shells to exist given by

This was based on a crush 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 nosotros rotate the region effectually the line $x=\text{−}k,$ where $k$ is some positive abiding. And 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, nosotros 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, nosotros 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 past ${x}_{i}^{*}.$ And then, the estimate volume of the shell is

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

The residue of the development proceeds as before, and nosotros come across that

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

We could as well rotate the region effectually other horizontal or vertical lines, such as a vertical line in the correct half aeroplane. 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 vanquish. To see how this works, consider the following case.

A Region of Revolution Revolved effectually a Line

For our last example in this department, let's look at the volume of a solid of revolution for which the region of revolution is bounded past the graphs of two functions.

A Region of Revolution Bounded by the Graphs of Two Functions

Which Method Should We Use?

Nosotros have studied several methods for finding the volume of a solid of revolution, but how practice we know which method to use? It often comes down to a selection of which integral is easiest to evaluate. (Figure) describes the dissimilar approaches for solids of revolution effectually the $x\text{-axis}.$ It'southward upward 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 bug and decide on the all-time approach to take for solving them.

Selecting the Best Method

Solution

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

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

If nosotros used the shell method instead, we would use functions of $y$ 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 peculiarly onerous, just since the shell method requires only one integral, and the integrand requires less simplification, we should probably go with the crush 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 aforementioned function. Therefore, nosotros can dismiss the method of shells. The solid has no crenel in the middle, then we can use the method of disks. Then

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

Select the all-time method to find the book of a solid of revolution generated by revolving the given region around the $x\text{-axis},$ and set up the integral to notice the volume (exercise 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; $V={\int }_{-1}^{1}\pi \left[{(2-{x}^{2})}^{2}-{({x}^{2})}^{2}\right]dx$

Cardinal 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 essentially 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 do, find the volume generated when the region between the two curves is rotated around the given axis. Utilize both the beat out method and the washer method. Utilize engineering to graph the functions and draw a typical slice by hand.

2. [T] Under the bend 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

iii. [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] Nether the bend of $y=2{x}^{3},x=0,\text{ and }x=2$ rotated effectually the $y\text{-axis}.$

vi. [T] Under the curve of $y=2{x}^{3},x=0,\text{ and }x=2$ rotated effectually 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 observe the volumes of the given solids. Annotation that the rotated regions lie betwixt the bend 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. $y=\frac{1}{x},x=1,\text{ and }x=100$

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

Solution

$\frac{2\pi }{3}$ units³

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

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 post-obit exercises, use shells to discover the volume generated by rotating the regions between the given curve and $y=0$ around the $x\text{-axis}.$

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

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

Solution

$\frac{32\pi }{5}$ units³

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

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

$\frac{-84\pi }{5}$ units^three

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

$\text{−}{e}^{\pi }{\pi }^{2}$ unitsⁱⁱⁱ

For the following exercises, observe the book 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 effectually the $y\text{-axis}.$

30. $y=\sqrt{x},x=0,\text{ and }x=1$ rotated around the line $x=2.$

Solution

$\frac{28\pi }{15}$ units³

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

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

Solution

$\frac{3\pi }{10}$ units^three

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

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 $y=2.$

For the post-obit exercises, utilize engineering science to graph the region. Make up one's mind which method you recollect would be easiest to utilise to calculate the volume generated when the office 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 $y\text{-axis}.$

xl. [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 post-obit exercises, apply the method of shells to approximate the volumes of some mutual objects, which are pictured in accompanying figures.

45. Apply the method of shells to find the volume of a sphere of radius $r.$

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 $r$ and height $h.$

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

$\frac{1}{3}\pi {r}^{2}h$ unitsⁱⁱⁱ

47.Use the method of shells to notice the volume of an ellipse $({x}^{2}\text{/}{a}^{2})+({y}^{2}\text{/}{b}^{2})=1$ rotated effectually 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 book of a cylinder with radius $r$ and height $h.$

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

Solution

$\pi {r}^{2}h$ unitsⁱⁱⁱ

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

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 past 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 contrary variable

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Source: https://opentextbc.ca/calculusv1openstax/chapter/volumes-of-revolution-cylindrical-shells/

Estimate the Volume if the Region Is Rotated About the Y Axis Again Use the Midpoint Rule With N 4

6.3 Volumes of Revolution: Cylindrical Shells

Learning Objectives

The Method of Cylindrical Shells

The Method of Cylindrical Shells 1

Solution

The Method of Cylindrical Shells two

Solution

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

A Region of Revolution Revolved effectually a Line

A Region of Revolution Bounded by the Graphs of Two Functions

Which Method Should We Use?

Selecting the Best Method

Solution

Solution

Cardinal Concepts

Key Equations

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Glossary

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