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; R is the outer radius and r the inner radius.
Recap
- The washer method works around any horizontal or vertical line, not just the coordinate axes.
- Slices are perpendicular to the axis: vertical (dx) for y = k, horizontal (dy) for x = h.
- Each radius is a distance from the axis: larger coordinate minus smaller coordinate.
- R belongs to the boundary farther from the axis; r to the nearer one. Decide with a test point.
- Changing the axis changes the radii, so the same region can give very different volumes.
Key relationships
| Axis | Where it is | Radius to the curve | Integrate in |
|---|---|---|---|
| y = k | above the region | k − f(x) | x |
| y = k | below the region | f(x) − k | x |
| x = h | right of the region | h − g(y) | y |
| x = h | left of the region | g(y) − h | y |
General form: V = π ∫ [R² − r²], with limits in the slice variable.
Assumptions behind the method
- The axis does not pass through the inside of the region.
- R ≥ r ≥ 0 across the interval. If the boundaries swap, split the integral.
- Around a vertical line, both curves are written as x in terms of y.
Mistakes to avoid
- “Top curve = R” when the axis is above the region. There, the lower curve is farther away.
- Subtracting k for an axis like y = −1. The distance is f(x) − (−1) = f(x) + 1.
- Ignoring the axis and using the plain curve values.
- Shifting the final answer instead of each radius.
- x-limits with dy (or y-limits with dx).
- (R − r)² instead of R² − r².
Quick self-check
- How far is the curve point at y = 2 from the axis y = 6? From the axis y = −3? (4; 5)
- How far is the point x = 3 from the axis x = −2? (3 − (−2) = 5)
- A vertical strip runs from y = 1 to y = 3 and is revolved around y = 4. Find R, r and the face area. (R = 4 − 1 = 3, r = 4 − 3 = 1, area 8π)
Next: practice questions.