Overview
Definite integration is used to find the area under a curve.
Definite integrals
Evaluate between limits a and b:
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
No constant of integration is needed (it cancels).
Area under a curve
The area between a curve and the x-axis from x = a to x = b is ∫ₐᵇ y dx.
- Areas below the x-axis come out negative — split the integral if the curve
crosses the axis.
- Area between two curves = ∫ (top curve − bottom curve) dx.
Worked example
∫₀² 3x² dx = [x³]₀² = 8 − 0 = 8.
Common mistakes
- Treating a negative result as the answer — area is positive, so take the modulus.
Exam tips
- Sketch the region first to see whether to split at the axis crossings.