Is this relationship purely an example of a mathematician searching for an equation that satisfies it, or is there a deeper, (possibly geometric), reason?
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See this paper for more examples – BAI May 13 '18 at 03:08
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1You can find answer for your question here https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80 – tien lee May 13 '18 at 03:14
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1As far as I know, this type of integral is related to investigating rational approximation of $\pi$, and indeed one of its variants is used for establishing finiteness of the irrationality measure of $\pi$. Perhaps this may be a good one to give a read. – Sangchul Lee May 13 '18 at 03:17
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There are integral of that nature with higher exponents that give an even better approximation of pi. I am sure they are on the internet somewhere – imranfat May 13 '18 at 03:28
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I'm pretty sure this question has been asked before on here... – Frank W May 13 '18 at 04:59
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2@FrankW. I know of a version on MO: https://mathoverflow.net/questions/67384/source-and-context-of-frac227-pi-int-01-x-x24-dx-1x2 (this is not to say there is none here; only that I know that one) – quid May 13 '18 at 11:10
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@FrankW. I have listed a few questions about this integral here in chat. Maybe somebody can find some other copies. – Martin Sleziak May 13 '18 at 11:31
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1Possible duplicate of Is there an integral that proves $\pi > 333/106$? – Xander Henderson May 13 '18 at 13:31
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An interpretation: https://math.stackexchange.com/questions/1662540/interpretation-of-frac227-pi – Jaume Oliver Lafont Jun 27 '18 at 03:20