Interesing, hard limit with sum, involving $\pi$

Interesing, hard limit with sum, involving $\pi$

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?