Washer Method in Calculus: Formula & Examples

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  • 0:02 The Washer Method
  • 1:21 Formula
  • 2:22 Washer Method with Lines
  • 3:20 Washer Method with Curves
  • 5:08 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you will learn how the washer method makes your calculus life a little easier by giving you an easy formula and procedure for finding the volume of various shapes created by simple (and not-so-simple) functions.

The Washer Method Defined

The washer method uses the shape of a washer to help us find the volume of a shape created by rotating two functions around the x-axis. A washer is a little piece of hardware that is used in construction and building projects. It is made out of metal and is in the shape of a flat doughnut.

Let's see how this washer relates to finding our volume using the washer method. The washer method is used when you have two functions where you want to find the volume between the functions. Let's start with a couple of simple functions: y = 1 and y = 2. This gives us two horizontal lines.

Now we want to find the volume between these two lines when we rotate them around the x-axis. What happens when we rotate each line around the x-axis? We get a circle from every point on the line. If we slice the resultant shape, the space between the two rotated lines will look like a washer.

Washer method

Can you see the washer shape in this illustration? No matter what kind of functions you have, the shape you get when you slice them will always be a washer of some size. Also, you will always have an outside function and an inside function. In this case, y = 2 is our outside function, and y = 1 is our inside function.


This washer shape helps us to find the formula for the volume. We know we can find the volume of this 3-D shape by integrating from a starting point to an ending point. The formula we need will provide us the area of each slice. Well, a washer is quite simply two circles of different size. You have a smaller circle inside a larger circle. The radius of each circle is provided by our functions.

The outer function is our outer radius, and the inside function is our inner radius. The formula for the area of a circle is pi * r^2. So the area of the washer is pi * R^2 - pi * r^2, where the R is the outer radius and r is the inner radius. Plugging the functions into our respective radii will give us a formula for the washer method we can use. Let's see what happens when we put this formula to use.

Washer Method with Lines

We will start with our first example with the two lines y = 1 and y = 2. Since these two lines go on indefinitely, we need to specify a starting point and an ending point. Our starting point will be x = 0, and our ending point will be x = 2.

Let's go through the process of evaluating the definite integral that is produced after plugging in all of the information into the formula. We've already established that our outer function is y = 2 and our inner function is y = 1. Let's plug this information into the formula and integrate.


So what we did was plug in our outer function and inner function where they belong and integrated it from the starting point to the ending point. After plugging in the functions, we were actually able to simplify the integral. At this point, we took the definite integral and evaluated it to get an answer of 6 pi. In calculus, pi is usually left as part of the answer and not multiplied out.

Washer Method with Curves

curved lines graph

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