Solid of Revolution

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 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.

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

Washer method example