JUMP TO TOPIC [show]
Washer Method – Definition, Formula, and Volume of Solids
Learning about the washer method gives us the ability to calculate the volume of different types of solid formed from two functions. We’ve learned how to calculate solids of revolution given one function before. In this article, we’ll explore how our method changes as we add one more function.
The washer is an extension of the disk method.
This time, we’re calculating the volume of solids formed by rotating the region between two curves or functions.
Since the washer method uses a similar process as the disk method, it’s important that you have understood the math behind the disk method. We’ll also be using our techniques in calculating the area between two curves, so make you remember the key concepts for this topic. Do you think you’re ready? Then let’s dive right in and understand what makes this method unique!
What is the washer method?
The washer method allows us to calculate the volume of solids of revolution using cylindrical disks with holes.
As we have mentioned, the washer method is an extension of the disk method. This technique is established so that we can also calculate for the volume of the solid returned by rotating the region bounded by two curves over the
Take a look at this image below – the disk was formed by rotating the rectangle on the left looks like an actual washer (it’s the flat ring we see in pipes of screws to ensure tightness). Hence, the name of this technique: the washer method.
Why do we have a gap between the axis of revolution as well as the rectangular cross-section? That’s because we’re looking at the region formed between two curves. We’ve learned in the past that we can simply subtract the areas under the functions’ curves to find the area between the two. The result is represented by the rectangular section with
We can use this expression to establish the mathematical definition of the washer method.
Washer method definition
Let’s say we have two non-negative and continuous functions,
The washer method allows us to calculate the volume of the solid formed by rotating
This means that the solid formed by rotating the region between curves is simply equal to the difference between their respective disks’ volumes. This also confirms what we know about the washer method – it is indeed an extension of the disk method.
Washer method formula
As with the disk method, we can also revolve the region between two curves about the
Here are two graphs with each having a region enclosed by two curves. The left graph is to be revolved around the
The variables in the formula for the volume will change depending on the axis of rotation. These are the two formulas defining the volume of the solid – one with respect to the
Horizontal Revolution | Vertical Revolution |
Use the appropriate formula based on the direction we want to revolve the region. Keep in mind that the lower and upper limits for the horizontal revolution will be based on the horizontal bounds (
How to use the washer method?
Once we understand how the washer method words, calculating the volume of the solid formed will then be straightforward. Here’s a quick guide to help you in setting up the graph and calculating the solid’s volume.
- Sketch the two functions’ curves to determine which of two lies above the other.
- Double-check if the curves meet the conditions of the washer method.
- When the bounds are not given, calculate for the functions’ points of intersection.
- Take note of the axis of revolution then apply the appropriate formula.
- Apply the appropriate definite integral properties and antiderivative
We’ve prepared some examples for you to work on and process the steps for the washer method. When you’re ready to work on more problems, head over to the next section as well.
Example 1
Determine the volume of the solid formed by rotating the region bounded by
Solution
Equate the two functions to find the points of intersection shared by the two functions.
Graph the two functions and use their points of intersection as a guide.
Since
Let’s begin by simplifying
This means that the solid formed by rotating the region between the curves,
Example 2
Calculate the volume of the solid formed by rotating the region bounded by
Solution
Since we’re rotating the graph about the
Graph the two curves and we’ll see that
This means that we’re rotating the region between
Here’s an illustration of how the region is rotated with respect to the
This means that the solid formed by revolving the region between the two curves about the
Practice Questions
1. Determine the volume of the solid formed by rotating the region bounded by
2. Calculate the volume of the solid formed by rotating the region bounded by
3. Determine the volume of the solid formed by rotating the region bounded by
4. Calculate the volume of the solid formed by rotating the region bounded by
5. Determine the volume of the solid formed by rotating the region bounded by
Answer Key
1. The solid has a volume of
2. The solid has a volume of
3. The solid has a volume of
4. The solid has a volume of
5. The solid has a volume of
Images/mathematical drawings are created with GeoGebra.