The Trapezium Rule

This lesson covers the trapezium rule, a method for numerical integration — approximating the value of a definite integral when exact integration is difficult or impossible. This is a required topic on the AQA A-Level Mathematics specification and is frequently examined.

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Why Numerical Integration?

Not all functions can be integrated analytically. For example, there is no elementary antiderivative of e^(x²), 1/ln x, or √(1 + x³). In such cases, we approximate the integral numerically.

The trapezium rule approximates the area under a curve by dividing it into trapeziums (trapezoids) rather than rectangles, giving a better approximation.

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The Formula

For n equally spaced strips on the interval [a, b], the strip width is:

h = (b − a)/n

The x-values are x₀ = a, x₁ = a + h, x₂ = a + 2h, ..., xₙ = b.

The corresponding y-values are y₀ = f(x₀), y₁ = f(x₁), ..., yₙ = f(xₙ).

The trapezium rule formula is:

∫(from a to b) f(x) dx ≈ h/2 × [y₀ + yₙ + 2(y₁ + y₂ + ... + yₙ₋₁)]

Or equivalently: