Simpson's rule

In Calculus, Simpson's Rule is a method of approximating integrals by using parabolas or other higher order polynomial expressions. It is named after its discoverer, Thomas Simpson.

Simpson's Rule

Specifically, Simpson's Rule states:

$\int _{a}^{b}f(x)dx\approx {{b-a} \over 6}\left[f(a)+4f({{a+b} \over 2})+f(b)\right]$

Composite Simpson's Rule

Simpson's Rule can be applied to segments of an integral, effectively making it a Riemann Integral.
If the interval $[a,b]$ is split into $N$ segments, with $N$ as an even number, there is a formula that can be used to solve for the integral.
Note that:

$x_{n}=x+nh$

$h={{b-a} \over N}$

As an extension of Simpson's Rule, we can find that:

$\int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{j=1}^{n/2-1}f(x_{2j})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]}$

The above formula can be rewritten as:

$\int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots +4f(x_{n-1})+f(x_{n}){\bigg ]}$

The formulas above do not work unless $N$ is even.

Other Versions of Simpson's Rule

There are a few other rules that Simpson proved. For example, Simpson's 3/8 Rule shows that cubic functions can be used to approximate integrals.