# Do definite Riemann integrals have a uniqe Riemann sums?

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?

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 $0$.