Composite Trapezoid Rule

This is the practical version of the Trapezoid Rule – we achieve improved accuracy by dividing a given curve into more intervals.

The total area, $I$, would be the summation of individual area:

$$I={\int }_{x_0}^{x_1}f(x)dx+{\int }_{x_1}^{x_2}f(x)dx+\cdots +{\int }_{x_{n-1}}^{x_n}f(x)dx$$

Using the trapezoid rule:

$$I=h\frac{f(x_0)+f(x_1)}{2}+h\frac{f(x_1)+f(x_2)}{2}+\cdots +h\frac{f(x_{n-1})+f(x_n)}{2}$$

In the equation above, $h$ is the width of the interval between two points. If we have $n$ equally spaced points, this would be:

$$h=\frac{b-a}{n}$$

For a cleaner formula, we can group the terms:

$$I=\frac{h}{2}\left(f(x_0)+2\sum _{i=1}^{n-1}f(x_i)+f(x_n)\right)$$

Or an even more general form:

$$I=\underset{\text{Width}}{\underbrace{(b-a)}}\underset{\text{Average height}}{\underbrace{\frac{f(x_0)+2{\sum }_{i=1}^{n-1}f(x_i)+f(x_n)}{2n}}}$$