Short on time? This page is the recap. For the picture, the reasoning and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: π ∫ (a to b) [radius]² dx means π times the definite integral of the squared radius from x = a to x = b.
Recap
- The disc method works about any horizontal or vertical line, not only the coordinate axes.
- Each slice is cut at right angles to the axis. Its radius is the distance from the axis to the curve.
- The region must touch the axis along its whole length. Otherwise the slices are washers (Topics 8.11 and 8.12).
- Spinning about y = k is the same as shifting the region and the axis down by k, then spinning about the x-axis.
- Volumes are in cubic units and never negative.
Key relationships
| Axis | Curve | Radius | Volume |
|---|---|---|---|
| y = k, curve below | y = f(x) | k − f(x) | π ∫ (a to b) [k − f(x)]² dx |
| y = k, curve above | y = f(x) | f(x) − k | π ∫ (a to b) [f(x) − k]² dx |
| x = h, curve to the left | x = g(y) | h − g(y) | π ∫ (c to d) [h − g(y)]² dy |
| x = h, curve to the right | x = g(y) | g(y) − h | π ∫ (c to d) [g(y) − h]² dy |
| Negative axis, e.g. y = −2 | y = f(x) above | f(x) + 2 | π ∫ [f(x) + 2]² dx |
Assumptions behind the method
- The axis is a boundary of the region on the whole interval.
- The solid is formed by one full turn.
- Limits are measured along the axis: x-limits for a horizontal axis, y-limits for a vertical axis.
Mistakes to avoid
- Measuring the radius from the x- or y-axis instead of from the axis of revolution.
- Sign slips with a negative axis: f(x) − (−2) = f(x) + 2.
- Squaring term by term: (1 − √x)² = 1 − 2√x + x, not 1 − x.
- dx for a vertical axis, or x-limits with dy.
- Using discs when the region does not reach the axis.
- Leaving out π, or using 2π.
Quick self-check
- The triangle bounded by y = x, y = 2 and the y-axis is spun about y = 2. Volume? (Radius 2 − x, so π ∫ (0 to 2) (2 − x)² dx = 8π/3, the volume of a cone of radius 2 and height 2.)
- A region is spun about x = 4. Which differential do you use? (dy, because the axis is vertical.)
- The curve y = f(x) is above the axis y = −3. What is the radius? (f(x) + 3)
Next: practice questions.