Year 12 Maths

Calculus: Integration

Learn antiderivatives, definite integrals, and how to calculate the area under a curve using the fundamental theorem of calculus.

Antiderivatives (Indefinite Integrals)

An antiderivative (or indefinite integral) reverses differentiation. If F′(x) = f(x), then F(x) is an antiderivative of f(x).

Power Rule for Integration

∫ xⁿ dx = xⁿ⁺¹ n + 1 + C (n ≠ −1)

C is the constant of integration — always include it for indefinite integrals.

∫ x³ dx

x⁴/4 + C

∫ 5 dx

5x + C

∫ x⁻² dx

−x⁻¹ + C = −1/x + C

Definite Integrals

A definite integral has upper and lower limits and gives a numerical value. It represents the signed area between the curve and the x-axis.

Fundamental Theorem of Calculus

∫_a^b f(x) dx = F(b) − F(a)

where F is any antiderivative of f.

Example: Evaluate ∫₁³ 2x dx

Step 1: Antiderivative of 2x is x²

Step 2: F(3) − F(1) = 3² − 1² = 9 − 1 = 8

Area Under a Curve

The definite integral gives the signed area between the curve and the x-axis. Areas below the x-axis are negative, so for the actual area, take the absolute value.

Above x-axis

Area = ∫_a^b f(x) dx (positive)

Below x-axis

Area = −∫_a^b f(x) dx (negate the result)

Example: Find the area between y = x² and the x-axis from x = 0 to x = 3

Area = ∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9 square units

Key Concepts Summary

●Integration is the reverse of differentiation. ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C.
●Always include + C for indefinite integrals.
●Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a).
●The definite integral gives signed area; take |value| for actual area below x-axis.
●Use initial conditions to find the constant C in particular solutions.