Numerical Integration of Exponential Tetration

Calculus Level 5

An exponential tetration is where xx is exponentiated by itself nn times.

For example,

4x=xxxx_{ }^{ 4 }{ x }={ x }^{ {\displaystyle x }^{ {\displaystyle x }^{\displaystyle x } } }

As nn approaches \infty, then for some 0<x<10<x<1, the function bifurcates at point BB, splitting into the upper branch where nn is even and the lower branch where nn is odd, even though in both cases, nn approaches \infty. Point AA is the origin (0,0)(0,0) and point CC is (0,1).(0,1). Let TT be the area of ABCABC as defined by the upper and lower branches, and the line x=0.x=0. What is the floor value 10000T?\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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