Short on time? This page is the recap. For the figures, explanations and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) A(x) dx means the definite integral of A(x) from x = a to x = b.
Recap
- A solid with known cross sections is a stack of thin slabs. One slab has volume about A(x) Δx.
- Adding the slabs and letting Δx → 0 gives V = ∫ (a to b) A(x) dx.
- The solid stands on a base region R. Each cross section rests on a segment across R.
- Perpendicular to the x-axis: segment length s(x) = top − bottom, limits are x-values, integrate dx.
- Perpendicular to the y-axis: segment length s(y) = right − left, limits are y-values, integrate dy.
- Volume is in cubic units.
Key relationships
| Slice | Area A | Volume |
|---|---|---|
| Square on side s | s² | ∫ [s(x)]² dx |
| Rectangle, height h(x) | s × h | ∫ s(x) h(x) dx |
| Rectangle, height = k × base | ks² | k ∫ [s(x)]² dx |
| Rectangle, constant height h | hs | h × (area of the base) |
| Any slice, same area A everywhere | A | A × length (a prism) |
Assumptions behind the method
- Each cross section is perpendicular to the stated axis and stands at right angles to the base.
- The area A(x) is continuous on [a, b], so the Riemann sums approach the integral.
- One formula for s works across the whole interval. If the top or bottom boundary changes, split the integral, as in Topic 8.4.
Mistakes to avoid
- Writing f² − g² instead of (f − g)². Find s first, then square it.
- Forgetting to square: ∫ s dx is the area of the base.
- Putting π into a square or rectangle area.
- Mixing x and y: a dy integral must be written entirely in y.
- Misreading the rectangle height, for example using s/2 instead of s × s/2.
- Rounding a calculator limit before integrating, or giving a decimal with no integral.
Quick self-check
- The base is the rectangle 0 ≤ x ≤ 4, 0 ≤ y ≤ 3. Slices perpendicular to the x-axis are squares. Find the volume. (36: each square has side 3, so A = 9 and V = 9 × 4. It is a 4 × 3 × 3 box.)
- The base is the region under y = √x, above the x-axis, for 0 ≤ x ≤ 4. Slices perpendicular to the x-axis are squares. Find the volume. (8: A = (√x)² = x and ∫ (0 to 4) x dx = 8.)
- A base region has area 5. Slices are rectangles of constant height 3. Find the volume. (15: V = ∫ 3s dx = 3 × 5.)
Next: practice questions.