Simpson’s rule

The elementary Newton-Cotes rule of second order, Simpson’s Rule, is known to all undergraduate calculus students – it’s one of the simplest, and best, methods of numerical integration (if the function is not too bizarre), especially in its compound form. But what is the easiest way of introducing it? In trawling through webpages and books, I’ve seen a myriad of approaches: Lagrangian interpolation, general formulas for the weights, and so on.

My favourite approach, which uses as little prior knowledge as possible, starts with the premise that Simpson’s rule is obtained by sampling the function at three equidistant points, fitting a second-order polynomial through those points, and integrating this polynomial.

To make the work easy for ourselves, we start by assuming the three equidistant points are -1, 0 and 1. Then the integral of a second-degree polynomial over this interval is

}

So, our three points are , , , and so, evaluating at the three values produces:

Then

so

which, from the integral computation above, produces

as the rule we require.