I am currently trying to solve a simple, yet difficult problem.
Suppose we are focusing on the unit square within the xy plane. There exist infinite integrable functions in this region. What is the average area of all the functions? Using basic logic I think the answer is:
1/2
This is because the maximum is 1 and the minimum is 0 and serves as an OK estimate for the answer to the problem.
I further tried to find some basic equations to describe these types of problems.
$J[y] = \int_0^1 y dx$
Is the functional of this problem. It maps the function to its area. The average should be like a "functional integral" or functional antiderivative of it. So if we could find a functional derivative that equals our existing one above we can take the difference in a similar sort of fundamental theorem of calculus. Where instead of b and a as our limits y=1 and y=0 as the limits.
Find $K[y] = \int_0^1 L(x, y, y')$ such that $\frac{\partial L}{\partial y} - \frac{\partial}{\partial x} \frac{\partial L}{\partial y'} = y$
Then the answer would be
$K[y=1] - K[y=0]$
The problem is that the functional derivative isn't unique. So there would be different numerical answers.
This is where I'm quite stuck, I haven't been able to solve the problem yet. Also, I've tried path integrals but there isn't a clear way to solve this problem with them. If you believe they are the answer please explain how to use the information of the region to evaluate the average.
Once again, thanks for all the help.