#simpson's rule

On my previous post, I gave an estimate for the natural log of two as:

\[\ln(2)\approx\frac{25}{36}\]

Here I will explain how I got it.

One definition of the natural log function, discovered in the early days of calculus before the number e was known to have anything to do with it, is this:

\[\ln(x)=\int\limits_{1}^{x}\frac{1}{t}dt\]

If the upper limit on the integral is 2, then we will be computing \(\ln(2)\).

\[\ln(2)=\int\limits_{1}^{2}\frac{1}{t}dt\]

Now, if we could evaluate this integral exactly (and get a rational number) then there would be no need to estimate the value of the integral. The best we can do is to find a nice function (one that we can integrate precisely) that approximates \(\ln(2)\) on the interval \([1\,2]\) and evaluate that instead. There are many ways to do this, but the one I will use is something called an interpolating parabola.

Here's what I'll do:

First: find a parabola that agrees with the function \(\frac{1}{x}\) at the points \(x=1\), \(x=\frac{3}{2}\), and \(x=2\). Parabolas are nice for this sort of thing. The equations for them are relatively easy to compute, and they tend to fit curves nicely, at least as long as the function doesn't do anything too crazy on the interval in question. For my purposes, a parabola is also one of the easiest types of curves to integrate, losing only to a straight line in this regard.

I won't bore you with the details of how I found the equation of the parabola, since if you have read this far and weren't scared off by the integral in the beginning, you probably already know how to do this. 

Here is the parabola I got:

\[y=\frac{1}{3}x^2-\frac{3}{2}x+\frac{13}{6}\]

Feel free to plug in the points in order to make sure that it agrees with \(\frac{1}{x}\) at the points \(1\,\frac{3}{2}\,\) and \(1\).

To find my estimate, all I have to do is integrate this function over the interval \([1\,2]\) instead of the original function \(\frac{1}{x}\).

\[\ln(2)=\int\limits_{1}^{2}\frac{1}{x}dx\approx\int\limits_{1}^{2}\frac{1}{3}x^2-\frac{3}{2}x+\frac{13}{6}dx\]

Evaluating this integral gives: \(\ln(2)\approx\frac{25}{36}\), and we're done!

It should be noted that this procedure can be streamlined in order to be executed more quickly. In addition, for better estimates we can split the interval into smaller pieces and use an interpolating parabola for each little piece. This procedure is known as Simpson's Method, or sometimes just Simpson's Rule.