Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function and $[a,b]$ an interval. Is there a way to estimate $$ \sum_{a \leq x_1 < .. < x_n \leq b} f(x_i) $$ as $n$ goes to infinity? If each sum was normalised then this is a like a Riemann sum... but since it's not, is there a way to approximate it? Any comments appreciated. thank u!
ps forgot to add (added after comments): I was thinking that with some kind of assumption on how $x_i$ is distributed on $[a,b]$, would it be possible to get an estimate with 'good' error term? based on the comments, perhaps one expects $$ \sum_{a \leq x_1 < .. < x_n \leq b} f(x_i) = n I + O(E) $$ where $I = \int_a^b f(x) dx$ and some error term $E$. Under what kind of assumptions can we make $E$ non-trivial? thank u!