I've been suggested to this site by some nice people at mathoverflow.net
Before I get started, let me tell you a little about myself. I’m a fourth year Mechanical Engineering student at the University of Michigan, Dearborn Campus. This said, I am not a mathematician, however I do know a bit about math, and I also enjoy doing math.
Please excuse any redundancies I may make and my “lack of rigor”.
The problem I’m working on now is an interesting one, and it goes as follows:
Suppose we have some unknown function of some real variable x, namely f(x), which is “well-behaved” on some known, finite interval, {a,b}. What is meant by “well-behaved” is that it is at least once differentiable, once integrate-able, smooth, and continuous on {a,b}.
Suppose further that we are given some finite area-under-the-curve A, and some finite arc-length S. Suppose finally that we are given f(x=a) and f(x=b).
The problem is this:
Given the above restraints on f(x), prove that there always exists some unique function f(x) that “fits the bill”.
For example:
Say f(x) = Sin(x), a = pi/2, and b = pi.
We know that:
f(a) = 1, f(b) = 0, A = 1, S = Sqrt[2]*EllipticE[1/2] ≈ 1.9101.
The problem in this case would be to prove that f(x) = Sin(x) is the only function that has the known area A, and known arc-length S.
Side Note: f(x) in this case could also be Cos(x - pi/2), however, this would not constitute two unique functions, f(x).
A corollary question could be posed if a proof to the first problem was given, and is as follows:
Given the same criterion for f(x) as before, devise a method of determining what f(x) would have to be to “fit the bill”. That is, like in our previous example, given: A = 1, S = 1.9101; devise a method of determining that f(x) must equal Sin(x).
Another Side Note:
I have realized (with help from mathoverflow.net) that f(x)=(x-1)^2(x+1) and f(-x) have the same Area and arc-length on [-1,1]. Making a potentially erroneous proposition, let us say that for all polynomials of nth order P(x), P(-x) will have the same Area and arc-length on any [-c,c].
This would prove difficult to prove analytically due to the "ugly" expression for arc-length.
While in general f(x) and f(-x) are unique functions, there must be some way of thinking about them as similar enough to "lump" them into one unique function... Just a thought
Thank you all. I look forward to any insight you can shed on the problem.