The Washer Method for Volumes of Revolution

What the Washer Method Is For

When a two-dimensional region is rotated around a line, it creates a three-dimensional solid. The washer method finds the volume of that solid by slicing it into thin pieces that are perpendicular to the axis of rotation.

Each slice produces a washer: a circular disk with a smaller circular hole removed from the middle. The outer circle has radius , and the inner hole has radius .

The key question is not simply "which formula do I memorize?" Instead, ask:

Once you know the outer and inner radii as functions of the variable of integration, the washer method is essentially just an application of the familiar area formula for a circle.

The Basic Idea

The area of a circle with radius is . A washer has an outer circle and an inner circle, so its cross-sectional area is:

Think of this as:

Then add up all of those infinitely thin washers with an integral.

The Washer Formula

If a region is revolved around an axis and the slices perpendicular to that axis have outer radius and inner radius , then:

The most important part of the formula is the subtraction:

Outer radius squared minus inner radius squared. The inner radius is not optional. If there is a hole in the middle of your solid, its area must be removed.

How to Identify the Radii

This is where most washer-method problems are actually solved. The integration itself is usually straightforward once the radii are correct.

1. Identify the axis of rotation. Is the region rotating around the x-axis, y-axis, or another horizontal or vertical line?
2. Determine the direction of your slices. Washers use slices perpendicular to the axis of rotation.
3. Find the outer radius. Measure the distance from the axis of rotation to the farther boundary of the region.
4. Find the inner radius. Measure the distance from the axis of rotation to the nearer boundary.
5. Find the bounds. Determine where the two relevant curves or boundaries meet.
6. Write the washer integral. Use .
7. Evaluate the integral and include cubic units in your final answer.

Vertical Slices: Rotation Around a Horizontal Axis

When the axis of rotation is horizontal, such as the x-axis, washers come from vertical slices. That means you normally integrate with respect to .

Suppose a region lies between the curves and , and is rotated around the x-axis. The vertical slice reaches from the lower curve to the upper curve.

If both curves are above the x-axis, then:

Notice that the radius is a vertical distance. Since the axis is the x-axis, the distance from the x-axis to a curve is simply its y-value.

When the Region Has a Hole

The washer method becomes especially useful when the region does not touch the axis of rotation.

For example, suppose a region lies between two curves and is rotated around the x-axis. A vertical slice might extend from to . When rotated, that slice creates a washer rather than a solid disk.

The distance from the x-axis to the upper curve is the outer radius, while the distance from the x-axis to the lower curve is the inner radius.

Rotation Around a Horizontal Line Other Than the x-Axis

The axis of rotation does not have to be the x-axis. Suppose the region is rotated around the horizontal line .

The radii are now distances from that line, not simply the y-values of the functions.

If the upper curve is and the lower curve is , with both below the axis , then the radius calculations must account for the vertical distance to that axis.

Horizontal Slices: Rotation Around a Vertical Axis

The exact same idea works when the axis of rotation is vertical. In that situation, washers come from horizontal slices, so you integrate with respect to .

For example, if a region is rotated around the y-axis and its horizontal slice runs from a left boundary to a right boundary , the radii are horizontal distances from the y-axis.

There is an important practical point here: if your curves are originally written as , you may need to solve for before you can use the washer method with respect to .

Washer Method vs. Disk Method

The disk method is really just a special case of the washer method.

If the region touches the axis of rotation, the inner radius is zero:

So the washer formula becomes:

That is the disk method.

A useful way to remember the relationship is:

Choosing the Correct Method

On an AP Calculus problem, don't automatically choose washers just because the problem involves volume. First look at the axis of rotation and the orientation of the slices.

• Washers/disks: slices are perpendicular to the axis of rotation.
• Shells: slices are parallel to the axis of rotation.

For a horizontal axis such as the x-axis, washers usually mean vertical slices and an integral with respect to .

For a vertical axis such as the y-axis, washers usually mean horizontal slices and an integral with respect to .

Finding the Bounds

Your bounds come from the interval over which the region exists. Often, this means finding where two curves intersect.

For example, if the region is bounded by and , solve:

The resulting x-values give the limits of integration when using vertical slices.

For horizontal slices, solve for the corresponding y-values instead.

A Reliable AP Exam Strategy

When you see a volume-of-revolution problem, use this checklist before doing any algebra.

1. Draw or visualize the region. You need to know which curve is farther from the axis.
2. Mark the axis of rotation.
3. Draw one representative slice. This is often the fastest way to identify the radii.
4. Decide whether the slice creates a disk or washer.
5. Write the radius expressions as distances.
6. Square both radii.
7. Subtract inner from outer.
8. Multiply by .
9. Integrate over the correct bounds.

A good AP Calculus habit is to write the radius definitions before writing the integral. For example:

Then substitute them into the formula. This makes your setup much easier to check.

Common Mistakes

Practice Problems