Finding the length of a section of a curve

This morning I was set a new problem, which I feel I will be working on for a fair while - what is the volume of a gas-filled pillowcase shape (two rectangles stuck together around the edges, and inflated to the maximum possible volume)? I already have my suspicions as to how the 3-D surface of a half-pillowcase shape could be modelled as a function - a 2-D function needs to be found which, for a given length of the curve (the length of one of the rectangles making up the pillowcase), maximises the area under it. Using a bit of intuition, this function should rise and fall very quickly at its ends, for example y = |√(x+5)|, and have a relatively high height inbetweentimes. This leads me to common naturally-occurring functions such as hyperbolae, catenaries or even quadratics! 

The first key bit of information needed is a method for calculating the length of a section of a curve. I have done this today, generalising the method up to an integral about halfway down; from there I took the quadratic example forward, since it turns out pretty neatly: