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Learn Index · Calculus Grade 11–12

Area under curves

The area between a curve and the x-axis is found by integrating.

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From slopes to areas

Derivatives measure slopes. Integrals measure area under a curve. The definite integral ∫ₐᵇ f(x) dx gives the area between y = f(x) and the x-axis from x = a to x = b, when f(x) ≥ 0.

Approximating with rectangles

Divide the interval into n strips of width Δx. Each strip is approximately a rectangle of height f(x).

Area ≈ Σ f(xᵢ) Δx

More, thinner rectangles give a better estimate. In the limit, the sum becomes the exact integral.

Exact area

Example: Area under y = x² + 1 from x = 0 to x = 3

∫₀³ (x² + 1) dx = [x³/3 + x]₀³ = 9 + 3 = 12 square units

Signed area

The definite integral gives signed area:

  • Above the x-axis: positive
  • Below the x-axis: negative

Example: ∫₀^(2π) sin x dx = 0 (equal areas above and below cancel).

Total area (all positive): split where the graph crosses the axis.

∫₀^π sin x dx + |∫_π^(2π) sin x dx| = 2 + 2 = 4

Area below the axis

Example: Area between y = x² − 4 and the x-axis from x = −2 to 2

  • The curve is below the axis here.
  • ∫₋₂² (x² − 4) dx = [x³/3 − 4x]₋₂² = (8/3 − 8) − (−8/3 + 8) = −32/3
  • Area = 32/3 ≈ 10.67 square units

Area between curves

Area between curves f (top) and g (bottom) from a to b:

A = ∫ₐᵇ [f(x) − g(x)] dx

Example: Area between y = x and y = x²

  1. Points of intersection: x = x² → x = 0, 1
  2. On [0, 1], x ≥ x² (top is y = x)
  3. A = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6

Applications

  • Distance travelled: area under a velocity–time graph
  • Total water flow: area under a flow-rate graph
  • Consumer surplus in economics
  • Work in physics: area under a force–distance graph

Worked examples

Example 1: Area under y = 2x from 0 to 4? ∫₀⁴ 2x dx = 16 (a triangle: ½ × 4 × 8 ✓).

Example 2: Area under y = eˣ from 0 to 1? e − 1 ≈ 1.718.

Example 3: Area between y = 4 and y = x² from −2 to 2? ∫₋₂² (4 − x²) dx = 32/3.

Common mistakes

  • Not splitting where the curve crosses the axis when total area is needed.
  • Subtracting curves in the wrong order (bottom − top).
  • Forgetting to find intersection points.

Try it yourself

  1. Area under y = x³ from 0 to 2.
  2. Area between y = x² and the x-axis from 1 to 3.
  3. Area between y = 2x and y = x².
  4. Total area between y = x and the x-axis from −1 to 1.

Answers: 1) 4 2) 26/3 3) 4/3 4) 1

Quick recap

  • The definite integral gives area under a curve (as signed area).
  • Split at x-intercepts for total area.
  • Area between curves = ∫ (top − bottom) between points of intersection.