Volumes of revolution are created when a plane region is rotated around a line, forming a three-dimensional solid. Calculus lets us find the exact volume by slicing the solid into many thin pieces and adding their volumes with an integral. The disk, washer, and shell methods are three ways to organize those slices.
Comparing them helps you choose the method that makes the setup simplest and least error-prone.
The disk and washer methods use slices perpendicular to the axis of rotation, so their cross sections are circles or rings. The shell method uses slices parallel to the axis of rotation, so each slice sweeps out a thin cylindrical shell. The best method depends on the shape of the region, the axis of rotation, and whether solving curves for x or y is easy.
A good decision guide starts by identifying the axis, deciding whether slices are perpendicular or parallel to it, and writing all radii, heights, and limits in one variable.
Key Facts
- Disk method: V = integral from a to b of pi[R(x)]^2 dx when rotating around a horizontal axis with no hole.
- Washer method: V = integral from a to b of pi([R(x)]^2 - [r(x)]^2) dx, where R is the outer radius and r is the inner radius.
- Shell method: V = integral from a to b of 2pi(radius)(height) dx or dy, using slices parallel to the axis of rotation.
- Use dx when slices are vertical and use dy when slices are horizontal.
- Perpendicular to the axis usually means disks or washers, while parallel to the axis usually means shells.
- Radii are distances to the axis of rotation, so they must be nonnegative expressions such as top minus axis, axis minus bottom, or outer minus inner distance.
Vocabulary
- Volume of revolution
- A volume of revolution is a three-dimensional solid formed by rotating a two-dimensional region around a line.
- Disk method
- The disk method finds volume by adding circular cross sections that have no central hole.
- Washer method
- The washer method finds volume by adding ring-shaped cross sections with an outer radius and an inner radius.
- Shell method
- The shell method finds volume by adding thin cylindrical shells whose volume is approximately circumference times height times thickness.
- Axis of rotation
- The axis of rotation is the line around which a region is revolved to create a solid.
Common Mistakes to Avoid
- Using the wrong slice direction, because disks and washers require slices perpendicular to the axis while shells require slices parallel to the axis.
- Forgetting to square both radii in the washer method, because the cross-sectional area is piR^2 - pir^2, not pi(R - r)^2.
- Measuring radius from the curve instead of from the axis of rotation, because a radius is always the distance between the axis and the slice or boundary curve.
- Mixing x-expressions with dy or y-expressions with dx, because the integrand and limits must all use the same variable.
Practice Questions
- 1 Find the volume formed by rotating the region under y = x^2 from x = 0 to x = 2 about the x-axis. Use the disk method.
- 2 Find the volume formed by rotating the region between y = 4 and y = x^2 from x = 0 to x = 2 about the x-axis. Use the washer method.
- 3 A region is bounded by y = x, y = 0, and x = 3, and it is rotated about the y-axis. Explain whether shells with dx or washers with dy would likely be simpler, and justify your choice.