The Trapezium Rule

This lesson covers the trapezium rule — a numerical method for approximating definite integrals — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to know the formula, apply it with a given number of strips, understand whether the approximation is an overestimate or underestimate, and know how to improve accuracy.

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Why Use Numerical Integration?

Some functions cannot be integrated analytically — that is, there is no formula for their antiderivative in terms of standard functions. For example:

• ∫ e^(-x²) dx
• ∫ √(1 + x³) dx
• ∫ sin(x²) dx

In these cases, we use numerical methods to approximate the value of the definite integral. The trapezium rule is the most common method at A-Level.

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The Idea Behind the Trapezium Rule

The area under a curve y = f(x) between x = a and x = b is approximated by dividing the region into n equal strips (intervals) and treating each strip as a trapezium rather than following the exact curve.

The area of a trapezium is:

Area = (1/2)(a + b) × h