Integration · Year 13

In Year 13 you learn: definite integrals and areas.

Integration reverses differentiation. If you know a gradient function, integrating gives back the original function plus a constant. Definite integrals add up infinitely many thin strips. So they give areas under curves, distances from velocities and totals from rates.

Key facts

• ∫ ax^n dx = (ax^(n+1))/(n + 1) + c for n ≠ −1. Also ∫ 1/x dx = ln|x| + c.
• An indefinite integral needs + c. A known point on the curve fixes c.
• Definite integral from a to b: F(b) − F(a), where F is any integral of f.
• The area between a curve and the x-axis is the definite integral. Area below the axis comes out negative, so split the integral at the roots.
• Integrating a rate gives a total: distance = ∫ v dt and water collected = ∫ (flow rate) dt.
• Trapezium rule with strips of width h: area ≈ h/2 (y₀ + yₙ + 2(y₁ + … + yₙ₋₁)).

Worked examples

Other years

• Integration · Year 12