The Big Idea: The Axis Does Not Have to Be an Axis
Most introductory volume problems rotate a region around the x-axis or y-axis. But AP Calculus problems often make the axis of rotation a different horizontal or vertical line, such as or .
The underlying methods do not change. You still use disks, washers, or shells. What changes is how you calculate the radius.
The most important idea is simple:
If the axis is , a point at height is a distance from the axis. If the axis is , a point at horizontal position is a distance from the axis.
Horizontal Lines of Rotation
A horizontal line has the form . When a region is rotated around a horizontal line, vertical distances are the natural radii for disks and washers.
For example, if the axis is and the curve is , then the radius from the axis to the curve is:
Notice that the radius is a distance, so it should be nonnegative. If the curve lies below the axis, the expression may initially look negative, but the geometric radius is the absolute distance.
Vertical Lines of Rotation
A vertical line has the form . When a region is rotated around a vertical line, horizontal distances become the natural radii.
For example, if the axis is and the curve is described by , the horizontal distance to the axis is:
Again, think geometrically: radius = distance from the axis to the piece of the region being rotated.
Choosing Disks, Washers, or Shells
The choice of method depends primarily on the direction of your slices.
- Disks: use slices perpendicular to the axis of rotation when the region reaches the axis and produces solid circular cross-sections.
- Washers: use slices perpendicular to the axis when there is a hole in the middle.
- Shells: use slices parallel to the axis. The shell radius is the distance from the slice to the axis.
The axis being horizontal or vertical does not automatically determine whether you use washers or shells. Instead, ask whether your slices are perpendicular or parallel to the axis.
The Most Useful Decision Rule
- Draw or visualize the axis of rotation.
- Decide whether vertical or horizontal slices are easier.
- If slices are perpendicular to the axis, think disks or washers.
- If slices are parallel to the axis, think shells.
- Calculate every radius as a distance to the axis.
- Check whether the outer radius and inner radius are reversed anywhere in the interval.
Worked Examples
Find the volume of the solid generated by rotating the region bounded by and the x-axis, for , around the line .
Find the volume generated when the region bounded by , , and is rotated around the line .
Find the volume generated when the region bounded by , , and is rotated around the line .
Find the volume generated when the region bounded by , , and is rotated around .
When the Axis Is Not at the Origin
This is where many AP Calculus students make a mistake. Suppose the axis is . The radius is not automatically .
You need the distance from the curve to the line:
Likewise, if the axis is , the horizontal distance from to the axis is:
Think of the axis as a physical line in the coordinate plane. The radius is measured from that line—not necessarily from zero.
Be Careful When the Region Crosses the Axis
If the region crosses the axis of rotation, the distance to the axis can change behavior. This is especially important with washers.
For example, suppose a region is rotated around . If part of the region lies above the axis and part lies below it, the geometry may change because different slices can produce different outer and inner radii.
In these situations, split the integral where the geometry changes.
Never assume that one washer formula works across an interval if the identity of the outer radius or inner radius changes.
A Powerful AP Exam Shortcut
When deciding between washers and shells, ask one question:
If the answer is "vertical," then vertical slices are perpendicular to a horizontal axis and parallel to a vertical axis. If the answer is "horizontal," the opposite is true.
This gives you a quick mental table:
Common Mistakes
If the axis is , the radius is not . The radius is . Always measure from the actual axis of rotation.
The orientation of the axis does not by itself determine the method. For a horizontal axis, vertical slices give washers, but horizontal slices give shells.
If the axis does not pass through the region, the solid may have a hole. Then you need , not simply .
For shells, the height is the length of the slice parallel to the axis. For vertical shells, this is usually a difference of y-values. For horizontal shells, it is usually a difference of x-values.
If the region crosses the axis or the nearest boundary changes, the radius formulas may change. Find the transition point and split the integral there.
Practice Problems
Find the volume generated by rotating the region bounded by , , and around the line .
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Find the volume generated by rotating the region under and above the x-axis for around .
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A region is bounded by , , and . Find the volume when it is rotated around .
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The region under and above the x-axis on is rotated around . Set up the volume integral using washers.
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