The **washer method** is a way to find the volume of objects of revolution. It’s a modification of the disc method for solid objects to allow for objects with holes. It’s called the “washer method” because the cross sections look like washers.

## How to Use the Washer Method

In order to use this method, the axis of rotation must be perpendicular to the radius of rotation. You may need to integrate y with respect to x, depending on the orientation of your solid. Watch the video for an example or read on below:

You work the method in two stages:

- Calculate the volume of the solid, ignoring the hole,
- Find the volume of the hole and then subtract it.

This can be accomplished with the following integral:

**Example question:** Find the volume of the solid of revolution bounded by y = x^{2} and y = x and rotated around the x-axis.

Step 1: **Create a graph to help you visualize the problem.** I used Desmos.com:

By looking at the intersections of the two functions, I can see that the two graphs form a leaf-like shape between x = 0 and x = 1. This is the shape we’re going to rotate. The interval from 0 to 1 is also the bounds of integration, which we’ll need to plug into the formula (i.e. from *a* to *b*).

Step 2: **Sketch the solid of revolution.** If you find visualizations tough, this is the most challenging step. One of the simplest ways to figure out the shape is to cut a piece of paper in the shape of the area (in this case, a leaf), then rotate it in your hands around a straw or stick. Or, you can do what I did: copy and paste the shape (in MS Paint), flipping it over the x-axis:

Step 3: Sketch in the washer. Here, you’ll need to decide if your washers will go along the x-axis or y-axis. This step is vital as it will determine which way you’re integrating (with respect to x, or y). For this example we’ll make washers traverse the x-axis:

Try integrating with respect to x first (it’s often the easier option).

Step 4: Set up your integral. From the steps above, we have the following items to plug into the formula:

- r
_{0}= x (the function y = x gives the radius for the outer washer) - r
_{i}= x^{2}(the function y = x^{2}gives the radius for the outer washer) - a = 0
- b = 1

Which gives:

Step 5: Calculate the integral using your graphing calculator or an online integral calculator. I used the calculator at IntegralCalculator.com for the following steps:

- Apply linearity:
- Applying the integral rule for power functions to solve the integrals:

Step 6: Calculate the definite integral. We’re integrating from 0 to 1, so:

- Plug in x = 0 into the integral from Step 5 and solve
- Plug in x = 0 into the integral and solve
- Subtract the two numbers.

π((1/3)-(1/5)) – π((0/3)-(0/5)) ≈ 0.419

**Note**: Remember that volumes must be positive, so if you subtract the wrong way around (i.e. F(a) – F(b) instead of F(b) – F(a)), you’ll get a negative amount.