When researching a Riemann sum it is defined as a way to find a definite integral like in the trapezium rule. The only problem is that the sum is defined in terms of f(x) but i haven't seen an example of an actual function inside a Riemann sum.
So the question is how would you spot a Riemann sum in a problem and write it as an indefinite integral?
A typical example might be
$$ \sum_{i=1}^n \frac{n}{n^2 + i^2} $$
You want to write this in the form from Ian's comment with $$ \frac{n}{n^2 + i^2} = f\left(a + (b-a)\frac{i}{n}\right) \frac{b-a}{n} $$
Multiplying by $n$, you want $$ \frac{n^2}{n^2 + i^2} = (b-a) f\left(a + (b-a)\frac{i}{n}\right)$$ The dependence on $i$ on the right side is only through $i/n$; we can get that by dividing numerator and denominator by $n^2$.
$$ \frac{1}{1 + (i/n)^2} = (b-a) f\left(a + (b-a)\frac{i}{n}\right) $$
We may as well take $a=0$ and $b=1$, and then this is $$ \frac{1}{1+(i/n)^2} = f(i/n)$$ i.e. $$ f(x) = \dfrac{1}{1+x^2}$$ So this sum could be written as a Riemann sum for $$ \int_0^1 \dfrac{1}{1+x^2}\; dx $$