Short on time? This page is the recap. For the diagram, explanations and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: π ∫ (a to b) [R(x)² − r(x)²] dx means π times the definite integral from x = a to x = b.
Recap
- Slice the region perpendicular to the axis. If the slice reaches the axis, it spins into a disc. If there is a gap, it spins into a washer (a ring).
- A washer’s face is a big circle minus a small circle: area = π(R² − r²).
- R = distance from the axis to the farther edge of the slice. r = distance to the nearer edge.
- Adding thin washers gives a Riemann sum, which becomes the volume integral.
- The disc method is the washer method with r = 0.
Key relationships
| Axis | Slices | Volume | Limits |
|---|---|---|---|
| x-axis | vertical, width dx | V = π ∫ (a to b) [R(x)² − r(x)²] dx | x-values |
| y-axis | horizontal, width dy | V = π ∫ (c to d) [R(y)² − r(y)²] dy | y-values |
For a region above the x-axis: R = top curve, r = bottom curve. For a region to the right of the y-axis: R = right curve, r = left curve (both written as x in terms of y).
Assumptions behind the method
- R ≥ r ≥ 0 across the whole interval. If the curves swap which one is farther from the axis, split the integral there.
- The region lies entirely on one side of the axis.
- Both radii use the same variable as the slice thickness.
Rounding (calculator questions)
- Store intersection values; do not retype rounded ones.
- Keep π inside the calculation and round only the final answer, usually to three decimal places.
Mistakes to avoid
- Writing (R − r)² instead of R² − r².
- Inner minus outer, which gives a negative value.
- x-limits in a dy integral (or y-limits in a dx integral).
- Using a disc when the region does not touch the axis.
- Forgetting π or putting it on one term only.
- Rounding a middle step, then carrying on.
Quick self-check
- A washer has R = 3 and r = 1. What is the area of its face? (π(9 − 1) = 8π)
- The rectangle between y = 1 and y = 3 for 0 ≤ x ≤ 2 is revolved around the x-axis. Find the volume. (π ∫ (0 to 2) [9 − 1] dx = 16π, a thick-walled tube)
- With R = 5 and r = 3, compare R² − r² with (R − r)². (16 and 4: the shortcut gives a quarter of the right face area)
Next: practice questions.