The previous post looked at the FWHM (full width at half maximum) technique for approximating integrals, applied to the normal distribution.

This post is analogous, looking at the 1/*e* heuristic for approximating integral. We will give an example in this post where the heuristic works well. The next post will give an example where the heuristic works poorly.

## The 1/*e* heuristic

In a function decays roughly exponentially, one way to approximate its integral is by the area of a rectangle with height equal to the maximum value of the integrand, and with base extending to the point where the integrand decreases by a factor of *e*, i.e. where its value is 1/*e* times less than the initial value. This technique is discussed in the book Street-fighting Mathematics.

This technique gives the exact value when integrating exp(-*x*) from 0 to infinity, and so it seems reasonable that it would give a good approximation for integrals that behave something like exp(-*x*).

## First example

For the first example we will look at the integral

for *x* > 0. We’re especially interested in large *x*.

To find where the integrand has dropped by a factor of *e* we need to solve

exp(1/*t* – t) = exp(1/*x* – *x* – 1)

for fixed *x*. Taking logs shows we need to solve

*t*² – *at* – 1 = 0

where

*a* = *x* + 1/*x* – *x* + 1.

Applying the quadratic formula and taking the positive solution gives

We approximate our integral by the area of a rectangle with height

exp(*x* + 1/*x*)

and width

(*a* + 1/*a*) – *x*.

The plot below show that not only is the absolute error small, as one would expect, the relative error is small as well and decreases rapidly as *x* increases.