Numerical Integration of Exponential Tetration

This problem is a little easier to solve if we looked at the function(s) in the other way. The curve as defined by the upper and lower branches, joining at point $B,$ can alternatively be described by the implicit equation

${ x }^{ { \displaystyle x }^{\displaystyle y } }-y=0$

where $x(y)$ arches smoothly from $(0,0)$ to ( $1,0)$ . Then any one of several possible numerical integration methods can be used, once we have an exact implicit equation. For example, given a valuie $0\le y\le 1$ , numerically determine $x$ using the secant method, and use it as a data point for numerical integration.

The critical point $B$ is

$\left( \dfrac { 1 }{ { e }^{ e } } ,\dfrac { 1 }{ e } \right)$

as shown in the given graphic

A more straightforward way to numerically integrate this is to use a loop to exponentially tetrate $0\le x\le { e }^{ -e }$ for some very large $2n$ and $2n+1$ , and use the difference as a data point for numerical integration. Thus, one can check the numerical results from both methods for confidence in final accuracy.

Note: The other critical point for the infinite exponential tetration is

$({ e }^{ \frac { 1 }{\displaystyle e } },e)$

which is the limit of convergence, i.e., for $x>{ e }^{ \frac { 1 }{ e } }$ it does not converge