**Contest:** Give an example of a continuous function $f$ that satisfies three conditions:

- $f(x) \geq 0$ on the interval $0\leq x\leq 1$;
- $f(0)=0$ and $f(1)=0$;
- the area bounded by the graph of $f$ and the $x$-axis between $x=0$ and $x=1$ is equal to $1$.

Compute the arc length, $L$, for the function $f$. The goal is to minimize $L$ given the three conditions above.

$\mathbf{\color{red}{\text{Contest results:}}}$ $$ \begin{array}{c|ll} \hline \text{Rank} & \text{User} & {} & {} & \text{Arc length} \\ \hline \text{1} & \text{robjohn $\blacklozenge$} & {} & {} & 2.78540 \\ \text{2} & \text{Glen O} & {} & {} & 2.78567 \\ \text{3} & \text{mickep} & {} & {} & 2.81108 \\ \text{4} & \text{mstrkrft} & {} & {} & 2.91946 \\ \text{5} & \text{MathNoob} & {} & {} & 3.00000 \\\hline \text{-} & \text{xanthousphoenix} & {} & {} & 2.78540 \\ \text{-} & \text{Narasimham} & {} & {} & 2.78 \\ \end{array}$$

**Original question after contest statement:** The contest question was inspired by this paper. Can anyone come up with a different entry than those listed in the table below?

$$ \begin{array}{c|ll} \hline \text{Rank} & \text{Function} & {} & {} & \text{Arc length} \\ \hline \text{1} & 1.10278[\sin(\pi x)]^{0.153764} & {} & {} & 2.78946 \\ \text{2} & (8/\pi)\sqrt{x-x^2} & {} & {} & 2.91902 \\ \text{3} & 1.716209468\sqrt{x}\,\mathrm{arccos}(x) & {} & {} & 2.91913 \\ \text{4} & (8/\pi)x\,\mathrm{arccos}(x) & {} & {} & 3.15180 \\ \text{5} & (15/4)x\sqrt{1-x} & {} & {} & 3.17617 \\ \text{6} & -4x\ln x & {} & {} & 3.21360 \\ \text{7} & 10x(1-\sqrt{x}) & {} & {} & 3.22108 \\ \text{8} & -6x^2+6x & {} & {} & 3.24903 \\ \text{9} & 9.1440276(2^x-x^2-1) & {} & {} & 3.25382 \\ \text{10} & (-12/5)(x^3+x^2-2x) & {} & {} & 3.27402 \\ \end{array}$$