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## Summary:

- It’s more a discussion about history of mathematics than about the actual mathematical problem. Moderators if you think it doesn’t fit here you may move this thread to “General Discussion”.

## Main Question or Discussion Point

Archimedes Riemann integral is one of the most elegant achievements in mathematics, I have a great admiration for it. Mr. Patrick Fitzpatrick commented on it as

Anybody want to share his/her feeling about rigorous definition of integrals or want to comment on it?

Please write your precious opinion about why it took 2000 years for integrals to come up in a rigorous manner?

Please during this discussion somebody please teach me what “error bound of ##1/500##” mean :-) does it mean that error of 1 digit in 500 digits ?

Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer and inner polygonal approximations of the object. It is attributed to the German mathematician Bernhard Riemann beacuse he in 1845 placed the approximation strategy of Archimedes in a general, rigorous mathematical context applicable to problems much more general than the computation of area. Riemann’s contribution was made more than 2000 years after Archimedes computed the area of parabolic and circular regions by the construction of ingenious elementary geometric devices. Archimedes calculated the area of the circle of radius 1 and provided accurate error bounds for his approximation; he calculated ##\pi## with an error bound of ##1/500##

Archimedes first devised and implemented the strategy to compute the area of nonpolygonal geometric objects by constructing outer and inner polygonal approximations of the object. It is attributed to the German mathematician Bernhard Riemann beacuse he in 1845 placed the approximation strategy of Archimedes in a general, rigorous mathematical context applicable to problems much more general than the computation of area. Riemann’s contribution was made more than 2000 years after Archimedes computed the area of parabolic and circular regions by the construction of ingenious elementary geometric devices. Archimedes calculated the area of the circle of radius 1 and provided accurate error bounds for his approximation; he calculated ##\pi## with an error bound of ##1/500##

Anybody want to share his/her feeling about rigorous definition of integrals or want to comment on it?

Please write your precious opinion about why it took 2000 years for integrals to come up in a rigorous manner?

Please during this discussion somebody please teach me what “error bound of ##1/500##” mean :-) does it mean that error of 1 digit in 500 digits ?