Volume with Disc Method: Revolving Around the x- or y-Axis
Choose perpendicular disc slices and measure a radius from a coordinate axis.
College Board Topic 8.9 · CED p. 155 (PDF p. 160) · CED effective Fall 2020; project course outline 2026–27.
Rotate a segment that reaches the axis and you get a disc. Its area is πr², so V=π∫r². About the x-axis, use vertical strips and dx; radius is vertical distance from y=0. About the y-axis, use horizontal strips and dy; radius is horizontal distance from x=0. The disc method requires slices perpendicular to the rotation axis.
For the region 0≤y≤√x on 0≤x≤4 rotated about the x-axis, r=√x, so V=π∫₀⁴ x dx=8π. Negative coordinates can still produce positive radii: radius is a distance to the axis, and its square is nonnegative. If the region does not reach the axis, there is a hole and you need a washer, not a filled disc.
Let's work through it
- For 0≤x≤√y and 0≤y≤4 rotated around the y-axis, use horizontal slices. Reason: They are perpendicular to the vertical axis.
- Radius is √y; V=π∫₀⁴ y dy=8π. Reason: Squaring the radial distance gives each disc’s area.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What direction are disc slices about the y-axis?
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Retrieval 1
What direction are disc slices about the y-axis?
Worked answer and reason
Horizontal, with dy.
They must be perpendicular to the axis.
Retrieval 2
What creates a washer instead of a disc?
Worked answer and reason
A positive gap between the region and the axis.
The rotation leaves a central hole.
Catch the tempting mistake
- Every rotation about the x-axis uses the graph value directly as radius.
- That works only when the axis is y=0 and the slice reaches it. Other axes require a distance; gaps require washers.
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Put it to work
- Unit 8 problem 19 · No calculator · analytic
- Unit 8 problem 20 · No calculator · analytic
For the teacher
- Use a paper strip or diagram to trace its rotation. Require axis, slice direction, radius, and bounds before evaluation.
Glossary and deeper reading · 1 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Washer method
The washer method uses circular slices with a central hole to compute volume. A slice with outer radius R and inner radius r has area π(R²−r²). For example, rotating the region 1≤y≤2 on a horizontal interval around the horizontal axis gives washer area 3π. Both radii are distances from the rotation axis and must be expressed using the same slice variable.