Using Riemann Upper sums to solve limits

[Gib Z]

I often see people use the Riemann definition of the integral to solve a certain limit-series computation, but they usually just skip a step that I can follow one way but not the other. Given the integral, I can see the limit-series that comes from it, but when trying to find the integral from the limit-series I have a problem.

For example, over the interval a=0 and b=1, $$\int^1_0 f(x) dx = \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n}\right)$$ using Riemann Upper sums.

So basically If required to somehow work out $$f(k/n) = something$$, so that I can re-express it in terms of f(x). So i guess the question is, does anyone know a general method to spot the function of k/n, as to find f(x) ?

 

[AiRAVATA]

I don't believe is any, rather than pure inspection.

In most examples I've seen, the sum can be worked to be on the most common form

$$\lim_{\Delta P\rightarrow 0}\sum_{n=0}^\infty f(x_i^*)(x_{i+1}-x_i),$$

where $P=\{x_0,...,x_n,...\}$ and $x_i^* \in (x_i,x_{i+1})$, so its easier to spot the $f$.

 

[Gib Z]

Damn I was hoping that wouldn't be the answer :( Thanks