 # Washer Method: Definition, Examples

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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. A thin, horizontal slice from the torus on the left is rotated around the y-axis. The result is a very thin washer on the right.

## 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:

1. Calculate the volume of the solid, ignoring the hole,
2. 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 = x2 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:

• r0 = x (the function y = x gives the radius for the outer washer)
• ri = x2 (the function y = x2 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:

1. Apply linearity: 2. 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:

1. Plug in x = 0 into the integral from Step 5 and solve
2. Plug in x = 0 into the integral and solve
3. 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.

CITE THIS AS:
Stephanie Glen. "Washer Method: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/washer-method-definition-examples/
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