Yesterday I was boring so I decided to derive formula for area of circle with integrals. Very good exercise, I think, because I forgot many, many things about integrals. So I started with:
$$\int\_{-r}^{r} \sqrt{r^2-x^2}dx$$ but I didn't have any clue how to count indefinite integral $\int\sqrt{r^2-x^2}dx$ (is it even possible? today I only found method for counting definite integral above with trigonometric substitution, but this does not apply in general), so I decided to use Riemann's theorem, since I only need to count definite integral. And everything was going well, till something extremely interesting happend. The last step I need to do is to find this limit:
$$\lim\_{n \to +\infty}\frac{1}{n}\sum\_{k=1}^{n}\sqrt{\frac{k}{n}\cdot\frac{n-k}{n}}$$
Surprisingly it is equal to $\frac{\pi}{8}$, and it is mindblowing ;-) but I only know that because I know formula for area of circle which I'm trying to derive. But without knowing it, is it possible to calculate this limit with relatively simple methods? I really, really want to to this in order to award my attempts. Can anybody help?