Continuous Function for Shortest Possible Arc Length Intersecting x-axis at zero and one

Question

What function f(x) yields the shortest arc length and satisfies the following conditions:

1. $f(0) = 0 $ and $f(1) = 0$
2. $f(x)\ge 0$ for $0 \le x \le 1$
3. The area under the graph of $f(x)$ from $0$ to $1$ is equal to $1$

Answer 1

It is an arc of circle, known as Dido's problem in variational calculus.

$$ R^2 ( \theta - \sin \theta \cos \theta ) =1;\, R \sin \theta = 1 ;$$

Numerical result : $ \theta \approx 61.3308^{0} ;\, R \approx 0.569863 .$

(Almost the circumcircle area on a unit side equilateral triangle, minor segment removed)