Volume with Disc Method: Revolving Around Other Axes
Measure disc radii as distances from the stated shifted axis, checking whether the region reaches it.
College Board Topic 8.10 · CED p. 156 (PDF p. 161) · CED effective Fall 2020; project course outline 2026–27.
A shifted axis changes the radius, not the disc formula. About y=k, a vertical slice that reaches the axis has radius |f(x)−k|. About x=h, a horizontal slice has radius |g(y)−h|. Square the distance and integrate with the matching variable.
For the region between y=1 and y=1+x on 0≤x≤2, rotating about y=1 gives radius x and volume π∫₀²x²dx=8π/3. Translating the whole picture together does not change that volume. A disc is valid only when the region reaches the rotation axis for each slice; if both boundaries lie away from it, identify inner and outer distances instead.
Let's work through it
- For the region −1≤x≤y−1 on 0≤y≤2 rotated around x=−1, choose horizontal slices. Reason: The axis is vertical, so perpendicular slices use dy.
- Radius is (y−1)−(−1)=y. Reason: Measure from the stated axis to the boundary.
- Volume is π∫₀² y²dy=8π/3. Reason: Each slice rotates into a disc.
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
About y=3, a boundary at y=5 has what radius?
See every recall question with its answer
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Retrieval 1
About y=3, a boundary at y=5 has what radius?
Worked answer and reason
2.
Radius is distance from 3 to 5.
Retrieval 2
Why is the region’s contact with the axis relevant?
Worked answer and reason
It decides whether there is a hole.
A gap rotates into a washer.
Catch the tempting mistake
- A shifted radius is always the graph value plus the axis coordinate.
- Use distance: subtract the coordinate of the axis and choose the appropriate nonnegative magnitude.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 8 problem 22 · No calculator · analytic
- Unit 8 problem 23 · No calculator · analytic
For the teacher
- Ask students to label the axis numerically on a sketch and draw a radius segment. Include a translated copy of an earlier example.
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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Disc method
The disc method finds a solid’s volume using circular slices perpendicular to the rotation axis. A slice with radius R has area πR². For example, rotating the region under y=x on [0,1] around the horizontal axis gives volume equal to the integral of πx². If the region leaves a hole around the axis, a washer rather than a full disc is needed.