Mathematica: High-precision solution for the area of a region

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High-precision solution for the area of a region
I hope you found a solution that worked for you :)
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Thanks to all those great people for their contributions!

(mathematica.stackexchange.com/users/41696/jie-zhu)Jie Zhu
(mathematica.stackexchange.com/users/4999/michael-e2)Michael E2
(mathematica.stackexchange.com/users/4999/michael-e2)Michael E2
A special thanks goes out to the (mathematica.stackexchange.com/questions/306004/high-precision-solution-for-the-area-of-a-region)Stackexchange community

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regions precision-and-accuracy area numerical-integration