Does this limit $lim_{n\to\infty}\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$ converge to $\pi/4$?

Question

While trying to find an approximate area of a quarter of a circle by splicing it into small rectangles and summing their areas I've reached a point where I have this formula: $$\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$$ Writing quick program and calculating the sum with n = 100, 100, 1000, 10k, 100k items suggest this sum converges to $\pi/4$, however I have no idea why. I've tried to search for known series converging to $\pi/4$ but nothing seems to resemble above formula.

Please note that in this question I'm not interested in what I was initially for, i.e. the area of a quarter of a circle. This was merely an exercise to show my nephew how we can approximate certain things.

Answer 1

The sum is nothing but a Riemann sum for $\int_0^{1}\sqrt{1-t^{2}}\, dt$. You can evaluate this by making the substitution $t=\sin\, \theta$ and using the formula $2 \cos ^{2}\, \theta =1+\cos\, (2\theta)$ and you will get $\pi /4$.

Answer 2

This directly leads
from fundamental theorem of calculus
(https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)
and
the definition of Reimannian sum
(https://en.wikipedia.org/wiki/Riemann_sum)