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# Solid of Revolution

Consider a stationary revolving door. The door itself is a rectangle. As people enter the revolving door, the door spins in a circle, pivoting around a center pole. Close your eyes and picture it. If the door filled the entire space as it revolves, what shape would the path of a revolving door create?

As the door spins, it creates a cylindrical shape. In general, if you rotate a rectangle around a fixed line, you will produce a cylinder. This cylinder is known as a solid of revolution because you obtained it by means of a rotation.

By rotating different objects in different ways you can produce different solids of revolution. Let's take a look!

## Solid of Revolution Definition

As mentioned before, by revolving a curve around a fixed line and filling it you obtain a solid. Since this solid is obtained by means of a revolution, it is called a solid of revolution.

A solid of revolution, also known as a volume of revolution, is a solid figure obtained from rotating a curve around a straight line. The line used as a reference for the rotation of the curve is known as the axis of revolution.

A solid of revolution needs to be visualized in the three-dimensional space, as it requires having volume. Start with a function $$f(x)$$ over an interval $$[a, b].$$

Figure 1. A function $$y=f(x)$$ graphed on the $$xy-$$plane

Next, rotate the curve about a given axis. This axis can be any, but usually, the $$x-$$axis is chosen in Calculus. You have to picture that the curve goes out of the screen!

Figure 2. The function has been rotated along the $$x-$$axis

By doing this, you obtain what is known as the surface of revolution.

Figure 3. The surface of revolution obtained from rotation $$f(x)$$ along the $$x-$$axis

Finally, you obtain the solid by filling what is inside the surface of the revolution. The result is a three-dimensional region.

Figure 4. The solid of revolution obtained from rotation $$f(x)$$ along the $$x-$$axis

Any other straight line can be used as an axis of revolution. For instance, you can use the $$y-$$axis, the line $$x=2,$$ or even a linear function, like $$y=x.$$ There are tons of possibilities!

## Volume of a Solid of Revolution

You can form two types of solids of revolution by revolving a curve around an axis: disks and washers. Here we will take a look at each one at a time.

### The Disk Method

The disk method is used when the axis of revolution is a boundary for the solid of revolution.

The disk method essentially slices the solid of revolution up into a series of flattened cylinders, or disks, hence the method's name. To find the volume of the entire solid, the volume of each disk is added together.

Figure 5. A disk obtained by rotating a segment of a function whose boundary includes the axis of rotation

### The Washer Method

When the axis of revolution is not a boundary for the solid of revolution, the washer method is used.

The washer method essentially slices the solid of revolution up into a series of flattened donut washers. A washer is essentially a disk with a hole in the middle or a disk within a disk!

The volume of each washer can be found by subtracting the volume of the inner disk from the volume of the outer disk. Then, to find the volume of the entire solid, the volume of each washer is added together.

Figure 6. A washer obtained by rotating a function whose boundary does not include the axis of rotation

In order to get the most accurate volume measurement, we should slice the solid up into infinitely many flattened washers. Need more information about this method? Check out our article about the Washer Method.

## Area of a Solid of Revolution

A surface of revolution is a little different. As its name suggests, it is something like a thin sheet or a skin.

The surface of revolution is the surface that bounds the solid of revolution.

Essentially, you can find a surface of revolution by rotating a curve around an axis, just like a solid of revolution. However, this figure is not filled up, it is a completely hollow mathematical object!

Figure 7. A surface of revolution is completely hollow

Please note that despite this may look like a washer, the surface of revolution is completely hollow. This means that a surface of revolution has no thickness, so it does not have a volume at all! A solid obtained through the washer method does have a thickness, so it has volume as well.

## Centroid of a Solid of Revolution

When studying solids of revolution you might come across the term centroid. This is mainly because the formula for finding the volume of a solid of revolution is very similar to the formula for finding the centroid of a thin plate, or lamina.

While it is possible to find the centroid of a solid of revolution, the calculation is far more complex, and it is out of the scope of this article.

## Formula for the Volume of a Solid of Revolution

In order to find the volume of a solid of revolution, you need to know first if it is obtained through the disk method or the washer method.

In the case of the disk method, the cross-section of a disk is a circle with an area of $$\pi r^{2}$$. If the axis of rotation is the $$x-$$axis, then the radius of each disk is given by the function, that is

$r=f(x).$

In order to add all the disks you need to integrate, so the formula for a solid of revolution obtained through the disk method is

\begin{align} V &=\int_a^b \pi \left(f(x)\right)^2\,\mathrm{d}x \\ &= \pi \int_a^b \left(f(x)\right)^2\,\mathrm{d}x. \end{align}

If your solid of revolution is obtained through the washer method instead, you need to remove the area of the inner function, so the formula is

\begin{align} V &= \int_a^b \pi \left( f(x) \right)^2\,\mathrm{d}x - \int_a^b \pi \left( g(x) \right)^2 \, \mathrm{d}x \\ &= \pi \int_a^b \left( \left( f(x) \right) ^2 - \left( g(x) \right)^2 \right) \, \mathrm{d}x. \end{align}

## Solid of revolution examples

Here you can take a look at some solids of revolution that can be obtained by different methods and with different axes of rotation. For information about how to calculate the volumes of these solids of revolution, please check out our articles about the Disk Method and the Washer Method.

### Disk method example

Consider the function

$y=x^2 \quad \text{for} \quad 0\leq x \leq 2.$

For the function given above:

1. Use the disk method to find the solid of revolution using the $$x-$$axis as the axis of rotation.
2. Use the disk method to find the solid of revolution using the $$y-$$axis as the axis of rotation.

Solution:

First, graph the function in the $$xy-$$plane.

Figure 8. Graph of the function in the $$xy-$$ plane

Since the solid of revolution depends on the axis of rotation, you should do each case one at a time.

• Use the disk method to find the solid of revolution using the $$x-$$axis as the axis of rotation.

Here, you need to rotate the function along the $$x-$$axis. Imagine that the curve comes out from the screen!

Figure 9. Rotation of the curve along the $$x-$$axis

The resulting region is now highlighted. Since this is a solid of revolution you also need to fill it up!

Figure 10. Solid of revolution obtained by rotating the function along the $$x-$$axis

This looks like a trumpet, right?

• Use the disk method to find the solid of revolution using the $$y-$$axis as the axis of rotation.

Now it's time to rotate the function along the $$y-$$axis. Once again, think of this as if it went inside and outside the screen circularly!

Figure 11. Rotation of the figure along the $$y-$$axis

This region is then highlighted and filled up.

Figure 12. Solid of revolution obtained by rotating the function along the $$y-$$axis

Now it looks like a parabolic antenna. Cool, isn't it?

### Washer method example

Consider the functions

$f(x)=-(x-2)^2+3 \quad \text{for} \quad 1\leq x\leq 3,$

and

$g(x)=-(x-2)^2+2 \quad \text{for} \quad 1\leq x\leq 3.$

Use the washer method to find the solid of revolution obtained by rotating the area bound between the two curves along the $$x-$$axis.

Solution:

As usual, begin by graphing both functions.

Figure 13. Graph of the functions in the $$xy-$$plane

Next, the functions are rotated along the $$x-$$axis producing two surfaces of revolution.

Figure 14. Surfaces of revolution obtained by rotation of the functions along the $$x-$$axis

To finish the washer method, the area bound between the two surfaces must be filled.

Figure 15. Solid of revolution obtained by rotating the area bound between the two functions

The resulting object, despite being hollow, is a solid of revolution. Think of it as if it was the thick skin of an unripe grapefruit!

## Solid of Revolution - Key takeaways

• A solid of revolution is a solid figure obtained from rotating a curve around a straight line named the axis of revolution.
• To obtain a solid of revolution of a function $$f(x)$$ over an interval $$[a, b]$$, you need to rotate the curve about a given axis (vertical or horizontal) which produces a three-dimensional region
• If the axis of rotation is a boundary of the curve, you can use the Disk Method to obtain the solid of revolution.
• If the axis of rotation is not a boundary of the curve, you need to use the Washer Method instead. The resulting solid of revolution will be hollow.
• A surface of revolution is the surface that bounds a solid of revolution. A surface of revolution has no thickness, hence it has no volume.

A solid of revolution is a solid figure obtained from rotating a curve around a straight line.

Consider a stationary revolving door. The door itself is a rectangle. As people enter the revolving door, the door spins in a circle, pivoting around a center pole. As the door spins, it creates a cylindrical shape. In general, if we rotate a rectangle around a fixed line, we produce a cylinder.

The formula for finding the volume of a solid of revolution is the integral of the cross-sectional area of the solid.

Any shape created by revolving a bounded curve around a horizontal or vertical line.

Solids of revolution are commonly used in engineering and manufacturing.

## Final Solid of Revolution Quiz

Question

What is a solid of revolution?

solid of revolution is a solid figure obtained from rotating a curve around a straight line.

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Question

What is a surface of revolution?

The surface of revolution is the surface which bounds the solid of revolution.

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Question

Given a function $$f(x),$$ how can you obtain a solid of revolution?

To obtain a solid of revolution starting with a function $$f(x)$$ over an interval $$[a,b],$$ rotate the curve about a given axis (vertical or horizontal), producing a three-dimensional region.

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Question

How can you obtain the volume of a solid of revolution using the disk method?

You need to slice the solid of revolution in disks, and then add the volume of all the disks.

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Question

Rotating a function around the $$z-$$axis produces disks (or washers) that are parallel to ____.

the $$xy-$$plane.

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Question

What are the two major methods for calculating the volume of a solid of revolution?

Disk and washer methods

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Question

What is the difference between the disk and washer methods?

The disk method sums the area of infinitely many circular disks while the washer method sums the area of infinitely many circular washers (a disk with a hole in the middle).

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Question

How do you know when to use the disk method?

The disk method should be used when the axis which the curve revolves around is also a boundary for the solid of revolution.

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Question

How do you know when to use the washer method?

When the axis which the curve revolves around is not a boundary for the solid of revolution, the washer method is used.

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Question

Rotating a function around the $$y-$$axis produces disks (or washers) that are parallel to ____.

the $$xz-$$plane.

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Question

Rotating a function around the $$x-$$axis produces disks (or washers) that are parallel to ____.

the $$yz-$$plane.

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Question

The line used as a reference for the rotation of a curve in a solid of revolution is called the axis of ____.

revolution.

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Question

True/False: The axis of revolution has to be either the x-axis or the y-axis.

False.

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Question

True/False: The axis of revolution can be the curve $$y=x^2.$$

False.

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Question

True/False: The axis of revolution can be the curve $$y=2x.$$

True.

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Question

True/False: A surface of revolution has no volume.

True.

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Question

True/False: The axis of revolution has to pass through the origin.

False.

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Question

A solid of revolution is also known as a(n) ____.

volume of revolution.

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Question

Suppose you need to find the solid of revolution obtained by rotating the region bound between $$f(x)=x^2,$$ the $$x-$$axis, and the line $$x=3.$$ The rotation is to be made around the x-axis. Which method do you need to use?

The disk method.

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Question

Suppose you need to find the solid of revolution obtained by rotating the region between $$f(x)=x^2,$$ $$g(x)=x^3,$$ and the line $$x=3.$$ The rotation is to be made around the $$x-$$axis. Which method do you need to use?

The washer method.

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