Pi (mathematical constant)/Proofs/An elementary proof that 22 over 7 exceeds π

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The rational number 22/7 is the most widely cited rational approximation to π. It is an early convergent in the continued fraction expansion of π. The following startling and elegant formula shows that it in fact exceeds π:

$0<\int _{0}^{1}{\frac {x^{4}(1-x)^{4}}{x^{2}+1}}\,dx={\frac {22}{7}}-\pi$

Evaluation of this integral requires only routine first-year calculus techniques, see Student level subpage.

What qualifies this as a proof is the fact that it demonstrates the result, but that is not its purpose in the present case, since the result can be derived by other means, including any method of computing π accurately. The formula naturally leads the reader to suspect that it is part of a larger pattern (see Lucas 2005 and Beukers 2000).

The problem of showing that this integral evaluates to 22/7 − π, perhaps despite its elementary nature, appeared in the 1969 Putnam Competition.

References

• Lucas, Stephen. "Integral proofs that 355/113 > π", Australian Mathematical Society Gazette, volume 32, number 4, pages 263–266, 2005.
• Beukers, Frits. "A rational approach to π". Nieuw Archief voor Wiskunde 2000, issue 4. (Online copy.)
• Dalzell, D. P. (1944). "On 22/7", Journal of the London Mathematical Society 19, pages 133–134.
• Dalzell, D. P. (1971). "On 22/7 and 355/113", Eureka; the Archimedeans' Journal, volume 34, pages 10–13.