Riemann Sum-esque limit involving Radicals

Question

Me and my friend were trying to evaluate the following limit: $$ I\equiv\lim_{N\to\infty}\sum_{k=0}^N\sqrt{\frac{1}{N^2}+\left(f\left(\frac{k+1}{N}\right)-f\left(\frac{k}{N}\right)\right)^2} $$ it was the last step in a problem we were trying to solve. It looks a lot like the formula for a Riemann sum mixed with the formula for the differential of arclength: $$ \int_0^1f(x)\text{d}x = \lim_{N\to\infty}\frac{1}{N}\sum_{k=0}^Nf\left(\frac kN\right) \\ \text{d}s = \sqrt{1+(f'(x))^2}\text{d}x $$ but we're not sure what the connection is (if there is one). Any ideas on how to evaluate $I$?

Answer 1

Hint: Write $~\dfrac1N~=~\sqrt{\dfrac1{N^2}}\quad$ and $\quad f\bigg(\dfrac{k+1}N\bigg)-f\bigg(\dfrac kN\bigg)~=~\dfrac{f\bigg(\dfrac{k+1}N\bigg)-f\bigg(\dfrac kN\bigg)}1~=~$

$~=~\dfrac{f\bigg(\dfrac{k+1}N\bigg)-f\bigg(\dfrac kN\bigg)}{(k+1)-k}~=~\dfrac1N\cdot\dfrac{f\bigg(\dfrac{k+1}N\bigg)-f\bigg(\dfrac kN\bigg)}{\dfrac{k+1}N-\dfrac kN},~$ where the last expression, for

$N\to\infty,$ looks suspiciously close to the definition of the derivative...