The 1/e heuristic

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

$\int_x^\infty exp\left(\frac{1}{t} - t \right) \, dt$

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

t² − at − 1 = 0

where

a = x + 1/x − x + 1.

Applying the quadratic formula and taking the positive solution gives

$t = \frac{a +\sqrt{a^2 + 4}}{2} \approx a + \frac{1}{a}$

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.