The Disc Method for Volumes of Revolution

What the Disc Method Is For

The disc method is one of the standard ways to find the volume of a solid created by rotating a region around an axis. The basic idea is surprisingly simple: imagine cutting the solid into extremely thin circular slices, find the area of each circular slice, and then add all of those areas using an integral.

If the region is rotated around the x-axis, vertical slices usually produce discs. If the region is rotated around the y-axis, horizontal slices may produce discs. The key is that each slice must be perpendicular to the axis of rotation.

A disc has no hole in the middle. That means the region being rotated must extend all the way from the axis of rotation to the outer boundary. If there is a gap between the region and the axis, the resulting cross-section is a washer instead.

The Basic Formula

Suppose a region is rotated around an axis and each perpendicular slice forms a disc with radius . The area of one circular cross-section is:

Because the radius usually changes from one slice to another, we integrate the cross-sectional area over the interval.

When the axis of rotation is the , and the radius is given by , the formula becomes:

The entire method can be remembered as: volume = integral of cross-sectional area.

How to Recognize a Disc Problem

Before writing an integral, ask three questions:

What is the axis of rotation? Is the region rotating around the x-axis, y-axis, or another horizontal or vertical line?
Which direction should I slice? Use slices perpendicular to the axis of rotation.
What is the radius of each disc? The radius is the distance from the axis of rotation to the outer edge of the region.

For rotation around the x-axis, vertical slices are perpendicular to the axis, so the thickness is and the integral is usually written with respect to .

For rotation around the y-axis, horizontal slices are perpendicular to the axis, so the thickness is and the integral is usually written with respect to .

The Most Important Idea: Identify the Radius

The hardest part of many volume problems is not the integration. It is identifying the correct radius.

If a region under is rotated around the x-axis, the vertical distance from the x-axis to the curve is simply . Therefore, the radius of each disc is .

Notice that the radius is a distance. This is why you should think geometrically before immediately writing an integral.

Worked Examples

Example 1: A Region Rotated Around the x-Axis

Find the volume of the solid formed by rotating the region bounded by and the x-axis from to about the x-axis.

Step 1: Identify the radius. The axis of rotation is the x-axis, so the radius is the vertical distance from the x-axis to the curve:

Step 2: Find the cross-sectional area. Each perpendicular slice produces a disc, so:

Step 3: Set up the volume integral.

Step 4: Integrate.

Result:

Example 2: A Square-Root Function

Find the volume generated when the region bounded by , the x-axis, and is rotated about the x-axis.

Step 1: Identify the radius. Since the axis is the x-axis, the radius is:

Step 2: Square the radius.

Step 3: Write the volume.

Step 4: Evaluate.

Result:

Example 3: Rotation Around the y-Axis

Consider the region bounded by , the y-axis, and . Find the volume generated when this region is rotated about the y-axis.

Step 1: Solve for x. Because we are rotating around the y-axis, horizontal slices are convenient. Starting with , solve for :

Step 2: Identify the radius. The radius is the horizontal distance from the y-axis to the curve:

Step 3: Write the disc area.

Step 4: Set up and evaluate the integral.

Result:

Example 4: Rotation Around a Horizontal Line Other Than the x-Axis

Find the volume generated by rotating the region bounded by , the x-axis, and about the horizontal line .

Step 1: Identify the radius. The radius is the distance between the curve and the axis :

Step 2: Square the radius.

Step 3: Write the volume integral.

Step 4: Evaluate.

Result:

Disc Method vs. Washer Method

The disc and washer methods are closely related. Both add up circular cross-sections. The difference is whether the cross-section has a hole in the center.

A disc has only an outer radius:

A washer has both an outer radius and an inner radius:

If the region touches the axis of rotation, there is no hole and the disc method applies. If the region does not touch the axis, the resulting cross-section has a hole and the washer method is required.

Don't automatically use washers

If the region reaches the axis of rotation, there is no inner radius. Writing a washer formula with an unnecessary inner radius can make an otherwise simple problem much harder.

The Slice Direction Matters

A useful geometric rule is:

For rotation around a horizontal axis, use vertical slices for discs or washers. For rotation around a vertical axis, use horizontal slices.

This is different from the shell method, where slices are typically parallel to the axis of rotation. Keeping these two ideas separate prevents many AP Calculus setup errors.

Common Mistakes

Forgetting to square the radius

The area of a circle is , not . The radius must be squared before you integrate.

Using the function as the radius without checking the axis

The function itself is not automatically the radius. The radius is the distance from the axis of rotation to the curve. If the axis is , for example, the radius may be rather than simply .

Using the wrong variable

If you use vertical slices, the thickness is and you integrate with respect to . If you use horizontal slices, the thickness is and you integrate with respect to .

Confusing discs and washers

Ask whether the region touches the axis of rotation. If it does, the cross-section can be a disc. If there is a gap between the region and the axis, you need a washer with an inner radius.

AP Calculus Strategy

On an AP free-response question, do not rush directly to the antiderivative. The setup is often the most important part.

Sketch the region mentally or on paper. Identify the axis of rotation.
Determine the slice direction. Perpendicular to the axis means discs or washers.
Write the radius as a distance. Do not guess it from the equation.
Square the radius.
Multiply by .
Use the correct bounds. The bounds correspond to the direction of integration.
Evaluate the definite integral.

If the problem asks only for the setup, stop after writing the correct definite integral. If it asks for the volume, evaluate it and include appropriate units.

Practice Problems

Question 1

Find the volume generated by rotating the region under from to about the x-axis.

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Question 2

Find the volume generated by rotating the region under from to about the x-axis.

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Question 3

Set up, but do not evaluate, the integral for the volume generated when the region under from to is rotated about the x-axis.

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Question 4

A region is rotated around the line . Explain why the radius is not simply .

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