how can i find a formulation for a curve that connects 2 fixed points on the horizontal axis, has a fixed arc length L, and encloses the maximum area??

If you are given 2 points and the arc length then I would imagine that there is only one solution making the "maximum area" the only possible area.

The x coordinate of the center of your arc will obviously be [math]\frac{x_1+x_2}{2}[/math]

 

The angle "a" of the arc can be expressed as:

 

[math]2(\Pi - tan^{-1}(\frac{|x_1-x_2|}{2y}))[/math] *where y is the y-coordinate of the center of the arc.

 

And along with this equality you should be able to solve it:

 

[math]ra = L[/math]

 

where:

 

[math]r = \sqrt{(x_1 - x_2)^2 + y^2}[/math]