The trapezium rule

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

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:

$$\text{area} = \frac{y(x_1) + y(x_2)}{2}\cdot \Delta x$$

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

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

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

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

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 $x = 1$ and $x = 5$.