The algorithm for these integration methods is rather simple, and the number of approximations perhaps unlimited!

• Choose a step size $h=(b-a)/N$ where $N$ is the number of steps and $a$ and $b$ the lower and upper limits of integration.
• With a given step length we rewrite the integral as

$$\begin{equation*} \int_a^bf(x) dx= \int_a^{a+h}f(x)dx + \int_{a+h}^{a+2h}f(x)dx+\dots \int_{b-h}^{b}f(x)dx. \end{equation*}$$

• The strategy then is to find a reliable polynomial approximation for $f(x)$ in the various intervals. Choosing a given approximation for $f(x)$, we obtain a specific approximation to the integral.
• With this approximation to $f(x)$ we perform the integration by computing the integrals over all subintervals.