# Integral from function approximations

I have some data which I cannot manage to model and fit with a known function, so let’s say that they are a sample from the unknow function $$f(x)$$, which look a sort of skewed bell-shaped distribution.

Given $$a,b \in \mathbb{R}$$ and $$\vec{w}$$ some parameters of the unknown function:

$$\int_{a}^{b} f(x,\vec{w}=\vec{w_1}) = A \,,$$

I would like to know if there is an approximate method that allow me to sampling or to have a function approximator of $$f(x,\vec{w}=\vec{w_2})$$ s.t. $$\int_{a}^{b} f(x,\vec{w}=\vec{w_2}) = A^c$$ where $$c \in \mathbb{R}$$. What I'm interested in are the tails of distribution $$f(x,\vec{w}=\vec{w_2})$$.

• Can you give us more information about "I cannot manage to model and fit with a known function"? Sep 20 at 12:53
• Are you just looking for any kind of function $g(x)$ that satisfies the second equation? How about $g(x)=\frac{A^c}{b-a}$? Sep 20 at 13:58
• Sorry I explained really badly, I reformulated the question. Sep 20 at 17:05
• I still don't understand. The function $f(x,w_2)$ is what it is. You can't choose what integral it has, the integral just is. Sep 22 at 0:12
• @WolfgangBangerth one could simply write $\int_{a}^{b} f(x,\vec{w}) = A^c$ in this case choosing the proper values of the parameters $\vec{w}$ I think one can surely "choose" the integral $A^c$ Sep 24 at 13:41