Question

Calculus of variations largest area given perimeter one fixed side

Answer 1

By reflecting the curve across the x-axis we can make it in to a closed loop, which doubles both the length and the area. Thus this is equivalent to the usual isoperimetric problem (maximize area inside a loop of fixed length) with the constraint of reflection symmetry. Since the circle is the unconstrained optimum and also has the reflection symmetry, it provides the solution to your problem: the curve is a semicircle.