Suppose we have the function $f(x) = x$. The definite integral of this function from 1 to a is:
$$ \int_1^a x dx = \frac{a^2-1}{2} $$
Now suppose we want to make $f(x)$ nonlinear, so it may take the form of something like $g(x) = x^2$. However, we want to constrain $g(x)$ so that its integral is the same as that of $f(x)$ despite the nonlinear term. Is there a general methodology we can use to make this transformation?
A one-variable definite integral $\int _a ^b f(x)dx$ is (Riemman's) the area between the $y=0$ axis and the $f(x)$ function, limited by $a \le x \le b$.
You can find infinite different $f(x)$ functions that have the same area within the same limits.
Think of a rectangle, can be the same area as a triangle, or an ellipse, etc.