The Riemann integral is defined in terms of Riemann sums of functions with respect to " tagged partitions " of an interval.
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His 1996 paper " Return to the Riemann Integral, " earned one of the Mathematical Association of America's highest honors in 1997.
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To be specific, we say that the Riemann integral of " f " equals " s " if the following condition holds:
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*PM : example of estimating a Riemann integral, id = 9236 new !-- WP guess : example of estimating a Riemann integral-- Status:
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*PM : example of estimating a Riemann integral, id = 9236 new !-- WP guess : example of estimating a Riemann integral-- Status:
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The Stratonovich integral can be defined in a manner similar to the Riemann integral, that is as a filtration of the Wiener process.
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For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
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However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral.
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The general statement can be proved by pairing up the terms in the series over " m " and converting the expression into a Riemann integral.
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Furthermore, every bounded function on a closed bounded interval has a Lebesgue integral and there are many functions with a Lebesgue integral that have no Riemann integral.
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