The disk and washer methods are calculus tools for finding the volume of a three dimensional solid formed by rotating a two dimensional region around a line. They matter because many real objects, from machine parts to containers, can be modeled as solids of revolution. Instead of adding up tiny rectangular areas, we add up many thin circular slices.
This turns a geometric picture into an integral that gives exact volume.
The disk method is used when each slice is a solid circle, so there is no hole in the middle. The washer method is used when each slice has an outer radius and an inner radius, creating a ring shaped cross section. In both cases, the key step is identifying the radius or radii as distances from the curve to the axis of rotation.
Once the radii are written as functions, the volume comes from integrating cross sectional area over the interval.
Understanding Volumes of Revolution - Disk and Washer Method
A useful way to build the setup is to imagine one thin slice of the original region before it turns. The slice must be perpendicular to the axis of rotation. A horizontal axis needs vertical slices, which usually means adding with respect to x.
A vertical axis needs horizontal slices, which usually means adding with respect to y. This choice is not just a convention. A slice parallel to the axis sweeps out a hollow tube rather than a disk or washer.
That situation belongs to the shell method. Drawing one representative slice and its rotated cross section often prevents choosing the wrong method.
The radius is always a distance, so it should never be negative. When the axis is the x-axis, the distance from a point at height y to the axis is the size of y. When the axis is a shifted line, students need to measure from that line rather than from zero.
For a region between two curves, label the curve farther from the axis as the outer edge and the curve nearer the axis as the inner edge. These labels can change across an interval.
If they do, split the integral at the point where the distances switch. A quick sketch with the axis clearly marked is more reliable than guessing from the names of the functions.
There is an important special case when the axis passes through a region. A slice may touch the axis, giving an inner radius of zero. That cross section is a disk.
If the axis cuts through the slice, the rotated shape can still be a disk, but its radius is the farthest distance from the axis. Using the difference of two squared radii without checking the picture can create a false hole.
This is one reason graphs matter in calculus. Algebra tells where curves meet, while the graph shows which parts of the region actually lie above, below, or across the rotation line.
The final integral adds volumes of thin slices, so the bounds must describe the direction in which the slices move. With vertical slices, use left and right endpoint values. With horizontal slices, use bottom and top values, then rewrite each boundary as x in terms of y when needed.
Units provide a useful error check. A radius has length units. Squaring it gives area units.
Multiplying by a tiny thickness gives volume units. In real settings, this reasoning models drilled holes in parts, bowls with empty interiors, bottles, pipes, and shapes made on a lathe. The model works best when the object is well approximated by smooth rotational symmetry.
Key Facts
- Disk method:
- Washer method:
- If rotating about the x-axis, then
- If rotating about , then radius =
- If integrating with respect to , then
- Cross sectional area for a washer: , and for a disk:
Vocabulary
- Solid of revolution
- A three dimensional shape formed by rotating a two dimensional region around a line.
- Axis of rotation
- The line about which a region is rotated to create a solid.
- Radius
- The distance from the axis of rotation to the outer edge of a disk or washer slice.
- Inner radius
- The distance from the axis of rotation to the inner edge of a washer, which creates the hole.
- Cross section
- A thin slice of the solid perpendicular to the direction of integration, used to build the volume integral.
Common Mistakes to Avoid
- Using the disk formula when there is a hole, which is wrong because a washer needs both an outer radius and an inner radius so the missing middle is subtracted.
- Forgetting that radius is a distance to the axis of rotation, which is wrong because the radius is not always just the function value and may require subtracting the axis location.
- Integrating with respect to x when the slices are naturally horizontal, which is wrong because the variable of integration must match how the cross sections are taken.
- Not squaring both radii in the washer formula, which is wrong because area depends on times radius squared, so subtracting unsquared radii gives the wrong value.
Practice Questions
- 1 Find the volume formed by rotating the region under from to about the -axis using the disk method.
- 2 Find the volume formed by rotating the region between and from to about the -axis using the washer method.
- 3 A region is rotated about the line and the graph lies partly above and partly below that line. Explain how you would determine the correct radius or radii for a disk or washer integral.