## Problem finding a continuous curved line given half of the starting curve...

I found out a problem I posted earlier is actually a calculus problem, after plotting values on an x,y axis and finding out it is a problem involving calculus and plotting a line on a graph.

I have three variables derived from any set 'x': x, e, and a.

After the set X1...XN (where all members of X are greater than or equal to zero, and all calculated y is greater than equal to zero) is established, I then determine the total of X, t and target of t, r; and r/t gives me 'e'. 'a' is determined by t/N. 'a' and 'e' are then constants at that moment, and so it t and r.

Then what I need is for all x,y given any set X - the sum of all y points = r exactly.

The problem is this: The y axis must curve sharply enough from x,y = 0,0 to x,y; x = a = x,xe. Then it must flatten out after that all the way to xN.

The actual problem and caveat is that for the entire set X, all resulting Y values must sum to total exactly r, with a sufficient curve from 0,0 to x,xe and then flatten for x > a, but the y values must sum to r exactly.

Now you can see that while I found an acceptable starting curve approximately just by doing numbers and theoretical realistic distributions, I still have not found the exact appropriate curve. I thought I needed to invert it for the rest, but I actually needed to continue it. Then I realized that I may not have the correct equasion even for the start of the curve, but it is approximately what I need....

The equasion that I have for the start of the curve which is approximately acceptable is for x <= a and needs to be something like the following, which determines the curved line from 0,0 to x = a. (x is any point along the x axis, and x is any member of the set), and this gives me 'y' (a point on the y axis):

x <= a, y = x2/a * e, or rather x2e/a, either way works out the same.

Then I just need to find the rest of the curve for x > a. I figure what I have done here is split up the curve into two parts up front, when it really needs to be a continuous curve all the way to 'xN'....

So what I mean by that is I may not actually have the real equasion for the actual curve. If I did, I should be able to calculate x > a in the same manner, or at least I figured I could, but it doesn't work that way either.... Now I guess you can see why I'm confused as to where I've been and gone with this thing...

Thanks for the help. It's appreciated.