Binomial analogue of Riemann sum for definite integral

I found this interesting relationship which is an analogue of the Riemann sum for definite integral.

$$ \lim_{n \to \infty}\frac{1}{n}\sum_{r = 1}^n f\Big(\frac{r}{n}\Big) = \int_{0}^{1}f(x) dx $$

$$ \lim_{n \to \infty}\frac{1}{2^n}\sum_{r = 1}^n {n \choose r}f\Big(\frac{r}{n}\Big) = f\Big(\frac{1}{2}\Big) $$

Application. $$ \lim_{n \to \infty}\frac{1}{2^n}\sum_{r = 1}^n {n \choose r}\Gamma\Big(\frac{r}{n}\Big) = \sqrt{\pi} $$

$$ \lim_{n \to \infty}\frac{1}{2^n}\sum_{r = 1}^n {n \choose r}\arcsin\Big(\frac{r}{n}\Big) = \frac{\pi}{6} $$

Question: Any reference to this in mathematics literature?

from Hot Weekly Questions - Mathematics Stack Exchange