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Trapezium Rule

The diagram below illustrates how we can approximate the area under the curve y=ln(x)y = ln(x).

By dividing the curve up into segments we can have created a set of trapezia.

The area of each trapezium is the average of the heights of its two parallel sides multiplied by its width:

area=y(x1)+y(x2)2⋅Δx\text{area} = \frac{y(x_1) + y(x_2)}{2}\cdot \Delta x

The total area under the curve is going to be the sum of the areas of these trapezia:

area=y(1)+y(2)2⋅Δx+y(2)+y(3)2⋅Δx+y(3)+y(4)2⋅Δx+y(4)+y(5)2⋅Δx\text{area} = \frac{y(1) + y(2)}{2}\cdot \Delta x + \frac{y(2) + y(3)}{2}\cdot \Delta x + \frac{y(3) + y(4)}{2}\cdot \Delta x + \frac{y(4) + y(5)}{2}\cdot \Delta x

We can factor out Δx2\frac{\Delta x}{2} to get:

area=Δx2(y(1)+y(2)+y(2)+y(3)+y(3)+y(4)+y(4)+y(5))=Δx2(y(1)+2y(2)+2y(3)+2(y(4)+y(5)))\begin{aligned} \text{area} &= \frac{\Delta x}{2}(y(1) + y(2) + y(2) + y(3) + y(3) + y(4) + y(4) + y(5)) \\ &= \frac{\Delta x}{2}(y(1) + 2y(2) + 2y(3) + 2(y(4) + y(5))) \end{aligned}

To generalise, we always have 2 of each of the middle terms and 1 of the inner and outer terms.

We could now evaluate this to get our approximation of the area under the curve between x=1x = 1 and x=5x = 5.

Tags: Mathematics