TITLE: Curve enclosing the maximum area
QUESTION [5 upvotes]: the curve of fixed length $l$ that joins the points $(0,0)$ and $(1,0)$ lies above the $x-axis$ and encloses the maximum area between itself and the $x-axis$, is a segment of 

A straight line
A parabola
An ellipse
A circle

I don't know exactly how to solve it but it seems Circle is the right Answer

REPLY [2 votes]: The above problem once again:
A curve of fixed length l lies above the x-axis, joins the points (0,0) and (1,0), and encloses the maximum area between itself and the x-axis is ---  a circle.
The constraints on l:
1) l>1 (obvious)
2) l=< π/2 
Let's look at constraint 2:
2) : As l approaches π/2 the centre of the circle moves up, from below the x-axis( negative y-values) along  a line parallel to.the x-axis through (1/2,0).
For l= π/2 the centre of the circle is at (1/2,0), maximal area is a semi circle.
Assume l>π/2 .
Problem: As before, this time with l>π/2.
Find the curve that encloses the maximal area between itself and the x-axis.
Comments welcome.