What is $\xi$ in the definition of the definite integral? $\sum_{n = 0}^{N-1}f(\xi_n)(x_{n+1} - x_{n}) $  and $\xi_n \in [x_n, x_{n+1}]$
So how I understand it: We could take equivalent distance between $x_n$ and $x_{n+1}$ and then we would get $x_n = a+\frac{n(a-b)}{N}$ and then define like this: $\sum_{n = 0}^{N-1}f(x_n)(x_{n+1} - x_{n}) $
But we cannot do this for every integral, so we do what is written above, we take arbitrary length of the interval $[x_n, x_{n+1}]$ .
However I do not understand why do we need $\xi_n$ to also be a number between the two point, and not just let's say $x_n$ like we used in equivalent distance formula.
 A: There are a number of variants of the usual definition. The Wikipedia page discusses them clearly, including your apparent suggested use of left Riemann sums throughout, which causes no problems since you can always refine tagged partitions to use left Riemann sums. It gives a simple counterexample to using both regularly-spaced sample points and left sums simultaneously--the indicator function $f(x) = 1$ if $x \in \mathbb{Q}$ and $0$ otherwise appears to be integrable over $[0, 1]$ with both restrictions, even though it is not integrable.
A: According to Wikipedia (edited to use $\xi$ rather than $t$):

One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, $\xi_i = x_i$ for all $i,$ and in a right-hand Riemann sum, $\xi_i = x_{i + 1}$ for all $i.$ Alone this restriction does not impose a problem: we can refine any
partition in a way that makes it a left-hand or right-hand sum by
subdividing it at each $\xi_i.$ In more formal language, the set of all
left-hand Riemann sums and the set of all right-hand Riemann sums is
cofinal in the set of all tagged partitions.

The real problem is if you limit yourself to left or right and limit your partitions to even partitions, $x_n=a+\frac{b-a}Nn.$ You can either limit your partitions to even partitions, or limit your $\xi_n$ to left or right, but not both.
Rather than pick one way, then prove the other is equivalent, we define with both together, then show this is equivalent to the two (pure left- or right-Riemann sums, or pure even partitions with arbitrary $\xi_n.$)
