Volumes of revolution are used to find the volume of a three-dimensional solid created by rotating a two-dimensional region around an axis. This cheat sheet helps students choose between disk, washer, and shell methods and set up the correct integral. It is useful when graphs, bounds, and axes of rotation make the radius or height change across an interval.
Strong setup skills prevent most errors in these problems.
The disk method uses circular slices with no hole, while the washer method subtracts an inner radius from an outer radius. The shell method uses cylindrical shells whose volume depends on radius and height. A good first step is deciding whether slices are perpendicular or parallel to the axis of rotation.
The basic formulas are , , and .
Key Facts
- The disk method is used when cross sections perpendicular to the axis of rotation are solid circles, with volume .
- The washer method is used when cross sections perpendicular to the axis of rotation have holes, with volume .
- For rotation around the -axis using vertical slices, the radius is usually a vertical distance such as .
- For rotation around the -axis using horizontal slices, the radius is usually a horizontal distance such as .
- The shell method uses slices parallel to the axis of rotation, with volume or .
- In the shell formula, is the distance from the slice to the axis of rotation and is the length of the slice across the region.
- When rotating around a line such as or , radii must be distances to that line, such as .
- Bounds of integration must match the variable of integration, so uses -values and uses -values.
Vocabulary
- Solid of revolution
- A three-dimensional solid formed by rotating a plane region around a line called an axis of rotation.
- Disk method
- A volume method that uses circular cross sections with radius and area .
- Washer method
- A volume method that uses ring-shaped cross sections with area .
- Shell method
- A volume method that adds thin cylindrical shells using or .
- Radius function
- A function that gives the distance from a slice to the axis of rotation.
- Axis of rotation
- The line around which a region is rotated to create a solid.
Common Mistakes to Avoid
- Using instead of in the washer method is wrong because circular area depends on the square of the radius.
- Choosing shell method with the wrong radius is wrong because the shell radius must be the distance from the slice to the axis of rotation, not the height of the region.
- Mixing bounds with functions of is wrong because the variable of integration must match the expressions and limits used in the integral.
- Forgetting to shift the radius when rotating around or is wrong because the radius is measured from the axis of rotation, not always from an axis.
- Subtracting functions in the wrong order is wrong because heights and radii must be nonnegative distances, such as .
Practice Questions
- 1 Find the volume when the region under from to is revolved around the -axis.
- 2 Set up and evaluate the washer integral for the region between and from to revolved around the -axis.
- 3 Use the shell method to set up the volume of the region under from to revolved around the -axis.
- 4 Explain how to decide whether disk, washer, or shell method is most efficient for a region rotated around a vertical line.
Understanding Volumes of Revolution (Disk, Washer, Shell)
The central skill is turning a flat picture into one small three-dimensional piece. Imagine choosing one thin strip of the region before it rotates. Its position determines its distance from the rotation line.
Its length determines how much of the region it covers. After rotation, a strip can form a circle-like slice or a hollow tube. Draw that one piece separately beside the graph.
Label every distance from the axis, not just from the horizontal or vertical coordinate line. A radius is always measured at a right angle to the axis.
For a shell, the height runs parallel to the axis. This geometric rule works for every rotation line.
Method choice is often about avoiding unnecessary algebra. A region described by a top curve and a bottom curve is usually easy to handle with vertical slices. If it rotates about a horizontal line, those slices naturally produce circular cross sections.
If the same region rotates about a vertical line, vertical shells may be simpler because their heights come directly from top minus bottom. A correct washer setup could require rewriting the curves in terms of y, which may be much harder.
Look at how the region is described before choosing a method. Choose the slice direction that keeps the radius, height, and bounds easiest to express.
A shifted rotation line creates many of the most common errors. When the axis is above, below, left, or right of the region, coordinate values are not automatically radii. The needed radius is the actual gap between a curve and that line.
If the region crosses the rotation line, the setup may need to be split into separate intervals. The curve that gives the outer edge can change from one interval to another. This matters especially for washers, where using the wrong outer radius gives a negative or incorrect cross-sectional area.
Shell radii must never be treated as negative distances. A quick sketch at several sample locations helps reveal when a change occurs.
Each volume integral is built from tiny approximate volumes. For a disk or washer, the small volume is cross-sectional area times a thin thickness. For a shell, it is circumference times height times a thin thickness.
Adding more and more thin pieces gives the exact volume. This explains why the final unit must be cubic units, such as cubic centimeters. These ideas appear in manufacturing when estimating material for pipes, containers, drilled parts, and turned objects made on a lathe.
Check a setup before calculating. Test one slice, confirm its radius and height, verify the bounds use the same variable as the thickness, and make sure the volume is zero where the region has no width or height.