I am looking for a function $f(x)$, defined on $[0,1]$ and depending on a parameter $\lambda \in (0,1)$, that approximately has the attached shape(s). One requirement is that $\int_0^1 f(x)dx = \lambda$. In my sketch I drew the function for different example values of $\lambda$.
Which functions take such a form? What would be their expression? Or which method could I use to find such a function?
I would be grateful for any help!
The function you are seeking can be easily described by my previous development of a generalized conics function called superconics, as described here. Alternatively, you may get away with using a subset of that which is the superparabola, which can be found in Wikipedia here. There are figures in both citations that you will see are what you are looking for.