Numerical Integration of Exponential Tetration

Calculus Level 5

An exponential tetration is where \(x\) is exponentiated by itself \(n\) times.

For example,

\[_{ }^{ 4 }{ x }={ x }^{ {\displaystyle x }^{ {\displaystyle x }^{\displaystyle x } } }\]

As \(n\) approaches \(\infty\), then for some \(0<x<1\), the function bifurcates at point \(B\), splitting into the upper branch where \(n\) is even and the lower branch where \(n\) is odd, even though in both cases, \(n\) approaches \(\infty\). Point \(A\) is the origin \((0,0)\) and point \(C\) is \((0,1).\) Let \(T\) be the area of \(ABC\) as defined by the upper and lower branches, and the line \(x=0.\) What is the floor value \(\left\lfloor 10000T \right\rfloor?\)

Note: Use of a computer and software is expected for numerical integration. The figure above is not drawn to scale.


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