# 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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