# Integration Using Simpson's Rule

## Key Questions

• Let us approximate the definite integral

${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$

by Simpson's Rule ${S}_{n}$.

First, split the interval $\left[a , b\right]$ into $n$ (even) equal subintervals:

$\left[{x}_{0} , {x}_{1}\right] , \left[{x}_{1} , {x}_{2}\right] , \left[{x}_{2} , {x}_{3}\right] , \ldots , \left[{x}_{n - 1} , {x}_{n}\right]$,

where $a = {x}_{0} < {x}_{1} < {x}_{2} < \cdots < {x}_{n} = b$.

Simpson's Rule ${S}_{n}$ can be found by

${S}_{n} = \left[f \left({x}_{0}\right) + 4 f \left({x}_{1}\right) + 2 f \left({x}_{2}\right) + 4 f \left({x}_{2}\right) + \cdots + 2 f \left({x}_{n - 2}\right) + 4 f \left({x}_{n - 1}\right) + f \left({x}_{n}\right)\right] \frac{b - a}{3 n}$.

I hope that this was helpful.

• Simpsn's Rule with $n = 10$ looks like this:
${S}_{10} = \frac{\Delta x}{3} \left[f \left(0\right) + 4 f \left(0.1\right) + 2 f \left(0.2\right) + 4 f \left(0.3\right) + 2 f \left(0.4\right) + 4 f \left(0.5\right) + 2 f \left(0.6\right) + 4 f \left(0.7\right) + 2 f \left(0.8\right) + 4 f \left(0.9\right) + f \left(1\right)\right]$,
where $\Delta x = \frac{b - a}{n} = \frac{1 - 0}{10} = 0.1$.

Would this get you going?

• Simpson's Rule will give you a better approximation of the integral than the other basic methods.

The other methods are Rectangular Approximation Method (RAM) - left, middle, and right; and the Trapezoidal Rule.

Numerical integration is used when we are given a set of data (evenly spaced on the independent variable) rather than an explicit function. This happens when data is collected from some measuring device.

Numerical integration is also used when it is difficult to find the antiderivative of the integrand.

And of course numerical integration is used on integrable functions to learn how it works.

Simpson's Rule is more accurate than the other methods because they use linear structures (rectangles and trapezoids) to approximate. Simpson's Rule uses quadratics (parabolas) to approximate. Most real-life functions are curves rather than lines, so Simpson's Rule gives the better result, unless the function that you are approximating is actually linear.

Simpson's Rule requires that the data set have an odd number of elements which gives you an even number of intervals. The other methods do not have this restriction.

Explains the subject in plain English can be found here: http://www.dummies.com/how-to/content/how-to-approximate-area-with-simpsons-rule.html