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
In order to get the exact volume, you need to slice the solid up into infinitely many disks. For more information about this method please reach out to our article about the Disk Method!
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.
Check out our article about Density and Center of Mass for more information about this topic!
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:
- Use the disk method to find the solid of revolution using the \(x-\)axis as the axis of rotation.
- 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.
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