Motivation for asking this question came from observing number of times some limit sums are turned into integrals.

Are there some limit sums that are not Riemann sum yet are equivalent to some integral?

Also is it possible to have two different Riemann sums evaluate to a same definite integral?


Riemann sums are not even close to being unique.

Given any partition $P$ of $[a,b]$ and any choice of $x$ in each of the subintervals, you can form a Riemann sum, and as long as $f$ is Riemann integrable and the size of the partition tends to $0$, the corresponding Riemann sum will converge to the value of the integral.

In particular, you don't have to split the interval into pieces of equal length, as long as you make sure that the length of the longest one goes to $0$.

  • $\begingroup$ So is there a name for the linearly partitioned Riemann sum that is used to shown to be equivalent to some integral? Where can i see some other examples of a limiting sum being equivalent to an integral. $\endgroup$ – Arjang Sep 22 '15 at 13:57
  • $\begingroup$ Here is one non-uniform (and one uniform) example: math.stackexchange.com/questions/589653 $\endgroup$ – mrf Sep 22 '15 at 13:58

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