The trapezium method allows you to approximate the area under a curve by breaking the curve up into and number of trapeziums whose areas can be easily calculated and then adding these areas up. This can be seen below.

A series of trapeziums can be used to approximate the area under a curve

When using the trapezium method the widths of the trapeziums used can be different, however it is often easier to calculate the total area (using the formula explained later) if all of the trapeziums are of equal width.

To improve the accuracy of the approximation you can use more trapeziums (the process of integration is simply allowing there to be an infinite number of trapeziums)

The area A of a trapezium with height (width when vertical in the approximation) h, and parallel sides a and b is given by

A=h(A+B)/2

To calculate the approximation we can let the x-values where each of the sides of the trapeziums touch the x-axis be x₀,x₁,x₂… and the y values where they cut the curve be y₀,y₁…

This means the the width of the first trapezium is x₁-x₀ which we will let equal d which is the same for all the trapeziums if we let their widths be equal. The parallel sides are of length y₀ and y₁ so the area A₀ is given by

A₀ = d(y₀+y₁)/2

The total area A under the curve is found by adding the areas of all of the trapeziums.

A = d(y₀+y₁)/2 + d(y₁+y₂)/2 + d(y₂+y₃)/2 + d(y₃+y₄)/2 +…….

However since d is a common factor it can brought outside in a bracket. Also all the y co-ordinates occur twice (in two consecutive trapeziums) apart from y₀ and the last y co-ordinate but all the terms are also halved this means that all but the first and last y values should be counted once and the first and last halved so we get the trapezium rule as follows

A = d(y₁+y₂+y₃ + … + y_n-1 +(y₀+y_n)/2)

where there are n trapeziums.