Short on time? This page is the recap. For explanations, the diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) [f(x) − g(x)] dx means the definite integral of f(x) − g(x) from x = a to x = b.
Recap
- Slice the region into thin vertical strips. Each strip has height top − bottom and width Δx.
- The Riemann sum Σ [f(xᵢ) − g(xᵢ)] Δx becomes the integral ∫ (a to b) [f(x) − g(x)] dx.
- Limits are the left and right edges: given vertical lines, or the x-coordinates where the curves meet.
- Top curve: test one x-value strictly between the limits.
- Top − bottom works above, below or across the x-axis. The x-axis is just the curve y = 0.
- If the top or the bottom boundary changes formula, split the integral at that x-value.
Key relationships
| Situation | Area integral |
|---|---|
| f above g on [a, b] | ∫ (a to b) [f(x) − g(x)] dx |
| f above the x-axis | ∫ (a to b) [f(x) − 0] dx |
| f below the x-axis | ∫ (a to b) [0 − f(x)] dx |
| Top changes from f to h at x = c | ∫ (a to c) [f(x) − g(x)] dx + ∫ (c to b) [h(x) − g(x)] dx |
| Curves meet at x = a and x = b | Solve f(x) = g(x) for a and b |
Assumptions behind the method
- Both functions are continuous on [a, b].
- One curve stays on top on each interval you integrate over. (If the curves cross inside, see Topic 8.6.)
- Each vertical line crosses the region in one piece, from one bottom curve to one top curve.
Mistakes to avoid
- Bottom minus top, giving a negative “area”.
- Using y-coordinates as limits in a dx integral.
- Splitting at the x-axis when it is not a boundary.
- Integrating only the top curve.
- Leaving out brackets or dx, so the integral is ambiguous.
- Rounding intersection points before evaluating on a calculator.
Quick self-check
- Find the area enclosed by y = x² and y = x. (They meet at x = 0 and x = 1; x is on top; ∫ (0 to 1) (x − x²) dx = 1/6)
- Find the area between y = 4 − x² and the x-axis. (Limits −2 and 2; ∫ (−2 to 2) (4 − x²) dx = 32/3)
- Your area integral gives −5. What went wrong? (With one curve on top throughout, you subtracted bottom − top. The area is 5.)
Next: practice questions.