Average length of chord of circle - definite integral
Solution 1:
Without loss of generality assume the circle is of radius 1 and divided by $n$ chords into $2n$ arcs. The required limit is then the Riemann sum $$\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}2\sin\frac{(2k+1)\pi}{2n}$$ which is thus equivalent to the integral $$\int_0^12\sin\pi x\,dx=\frac4\pi$$ Therefore, for a circle of radius $a$, the average length is $\frac{4a}\pi$.