Consider all continuous, (piecewise) differentiable functions such that
3) The area bounded by and the x-axis, is 1.
What is the smallest possible arc-length of from 0 to 1?
I believe that this is an open question, and would be interested in hearing about any thoughts that you have. You might need to apply numerical integration techniques to calculate arc length using calculus
By the iso-perimetric theorem, we know that the curve with the smallest perimeter which has an area of 1, is the circle with radius . This has a perimeter of , and hence the smallest possible arc-length is bounded below by . Of course, this doesn't yield a function, nor does it have a straight line segment, so it doesn't qualify. Can you improve this bound?
For an easy upper bound, we can take the square, which has a perimeter of 3. Of course, this strictly speaking isn't a function, but we could approximate it with piecewise linear functions. Can you improve this bound?
We have shown that . Can you tighten this further? To within