Volume with Washer Method: Revolving Around Other Axes
Measure both washer radii from the same shifted axis and keep the slice variable and bounds consistent.
College Board Topic 8.12 · CED p. 158 (PDF p. 163) · CED effective Fall 2020; project course outline 2026–27.
For a shifted axis, measure outer and inner distances from that same line, then use π∫(R²−r²). About y=k, use vertical slices; about x=h, use horizontal slices. Draw a short radius segment to each boundary. The outer radius may come from the lower or left boundary if the axis is above or to the right of the region.
For x≤y≤2 on 0≤x≤2 rotated about y=3, the farther boundary is y=x, so R=3−x and r=1. The volume is π∫₀²[(3−x)²−1]dx=20π/3. A calculator can evaluate a correct integral, but it cannot choose the radii for you. Keep exact π and other constants until the final requested rounding.
Let's work through it
- Rotate y²≤x≤y, 0≤y≤1, about x=−1. Use R=y+1 and r=y²+1. Reason: The right boundary is farther from the vertical axis left of the region.
- V=π∫₀¹[(y+1)²−(y²+1)²]dy. Reason: The washer formula uses distances squared separately.
- Expand 2y−y²−y⁴ and integrate to obtain 7π/15. Reason: Polynomial integration evaluates the volume without early rounding.
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
If an axis lies above the region, which boundary is farther: top or bottom?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
If an axis lies above the region, which boundary is farther: top or bottom?
Worked answer and reason
The bottom boundary.
Outer radius is the greater distance from the axis.
Retrieval 2
When should a volume’s decimal approximation be rounded?
Worked answer and reason
At the final reported value.
Intermediate rounding can change the final result.
Catch the tempting mistake
- Outer minus inner radius can be squared once at the end.
- Square each radius separately, then subtract. The two expressions describe different areas.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 8 problem 23 · No calculator · analytic
For the teacher
- Give an axis above the region and an axis to its left. Have students complete a radius-and-variable table, then explain why the answer is positive and has cubic units.
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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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.