Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$

Question

Let we have a continuous function $f(x)$ in the interval $ [ a,b ] $

Does there exist any relationship between its integral and summation of function-values defined at the integers between $a$ and $b$.

i-e Relationship between $\int_a^b f(x) dx$ and $\sum_{i= \lceil a\rceil}^{\lfloor b\rfloor} f(i)$ ?

For instance we have an integral test for infinite series which if positive and decreasing, then both integral and summation converges. But what can be inferred about the partial sum of series (not necessarily decreasing) if we know the integral between some finite limits ?

Answer 1

Yes. Euler-McLaurin's formulas completely describe this kind of relations.