This is a follow-up question of: I need to define a family (one parameter) of monotonic curves
$x_0$, $y_0$ are known constants. What's the integral of this function?
$f(x, a) = \frac{y_0}{1+\frac{(1-a)x}{a(x_0-x)}}$
A few words on the function itself. For $x_0 = 1, y_0 = 1$, it looks the following diagram. We get a different curve for a different value of $a$, the straight line is when $a = 0.5$, the curves below are have $a < 0.5$ and the ones above have $a > 0.5$.
Basically I need a function that I give it the total area to be covered, and the result is a value of $a$ that produces a curve covering that area. The valid areas are in $[0.0, x_0 * y_0]$. Some examples of this function $g$:
$g(0) = 0.0$ (when $a = 0.0$, the curve has zero area)
$g(x_0 * y_0) = 1.0$ (when $a = 1.0$, the curve has maximum area, the whole rectangle)
$g(0.5 * x_0 * y_0) = 0.5$ (when $a = 0.5$, the curve is a triangle)
I know, from tinkering with it, that the sought function has a sigmoidal shape like this:
(Horizontal axis is the value $a$, from $0.0$ to $1.0$, vertical axis is the area $R$, R for aRea)
My understanding is that I would need to integrate $f(x, a)$ (dx), get a closed formula involving $a, x_0, y_0, R$, and solve for $a$.
I admit that I was probably sleeping/absent when they taught me integration techniques in school and university. Any help much, much appreciated. I would also give a shot to wolfram alpha which has integration examples, but the problem is I have a couple of extra (fixed) parameters, and the syntax of it gives me trouble (or it only supports one symbolic name per function -- but that's unlikely)