**Simpson's Rule - Approximating Definite Integrals **

This is the last method we're going to take a look at and in this case we will once again divide up the interval [a, b] into n subintervals. Though, unlike the preceding two methods we want to require that n be even. The cause for this will be obvious in a bit. The width of every subinterval is,

Δx = b - a / n

In the Trapezoid Rule (explain earlier) we approximated the curve along with a straight line. For this Rule (Simpson's Rule) we are going to approximate the function along with a quadratic and we're going to need that the quadratic agree with three of the points from our subintervals. Below is a drawing of this using n = 6. Every approximation is colored in a different way thus we can see how they actually work.

Note: In fact each approximation covers two of the subintervals. This is the cause for requiring n to be even. A few approximations look much more like a line after that a quadratic, but they really are quadratics. As well note that some of the approximations do a better job as compared to others. It can be illustrated that the area under the approximation on the intervals [x_{i -1}, x_{i}] and [x_{i} , x_{i+1}] Δ is like this:

A_{i} = Δx / 3 (f(x_{i-1})+4f(x_{i}) + f (x_{i+1}))

If we make use of n subintervals the integral is then approximately,

∫^{b}_{a} f (x) dx ≈ Δx / 3 (f(x_{0}) + 4f (x_{1}) + f (x_{2}) + Δx / 3 (f (x_{2}) + 4f (x_{3}) + f (x_{4})) + ....+ Δx / 3 (f (x_{n}-_{2}) + 4f (x_{n-1}) + f (x_{n}))

On simplifying we reach at the general Simpson's Rule.

∫_{a}^{b} f (x) dx ≈ Δx / 3 [(f(x_{0}) + 4f (x_{1}) + 2f (x_{2}) .... + 2f (x_{n-2}) + 4f (x_{n-1}) + f(x_{n})]

In the above case notice that all the function evaluations at points along with odd subscripts are multiplied by 4 and every function evaluations at points with even subscripts (apart from for the first and last) are multiplied by 2. If you can keep in mind this, this is a quite easy rule to remember.