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Math Help - Elegant Meaning of an Integral

  1. #1
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    Elegant Meaning of an Integral

    For the formalists out there I have been troubled by the meaning of,
    \int 2x dx=x^2+C
    Because what is that supposed to represent?
    It is more appropriate to think of an integral as a set of functions rather than a function.

    So here is my version, tell me if you like it.

    Let \mathcal{R} represent the set of all real functions that are differenciable on some open interval. Let R represent the set of all functions. *)

    We can view the sets \mathcal{R} and R as groups under functional addition.

    Then we can define the derivative as a group homomorphism,
    d:\mathcal{R}\to R

    I think the beauty in this is defining what an integral is. Given any function f\in R we define,
    \int f = d^{-1}[f]

    *)Since I am not a set theorist (noob ) I cannot tell whether such a set can exist.

    Note: It was not necessary for me to mention a group homomorphism I just was thinking whether the kernel of the derivative will be of any use. I do not see of one.
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    TD!
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    In my opinion, the 'confusion' lies in the fact that the integral sign (∫) is used in two very different ways.
    For many people, this leads to the conclusion that there are two 'types' of integrals, which is not right.

    The actual 'integral' is a definite integral. Referring to the (relatively easy to understand) Riemann-integral, it is clearly defined in such a way that it returns what our understanding of an area is. The Riemann sum represents an approximation of the area under f(x) which we're integrating, the limit (the integral) is by definition the area.

    Now, when dropping the limits of integration, we get what's been given the name indefinite integral. It should be clear that this isn't another type of integral, it is merely (a perhaps unfortunately chosen) notation for a concept we can perfectly define without 'integration theory'. Given a function f(x) ([a,b] to R), we call a differentiable function F(x) ([a,b] to R) a primitive function of f(x), if F(x)' = f(x), for all x in [a,b].

    Because of historical reasons, we introduced the notation:

    \int {f\left( x \right)dx}  = F\left( x \right) + C (*)

    With F(x) defined as above. But we can also 'define' the indefinite integral, as a definite integral:

    \int {f\left( x \right)dx}  = \int\limits_a^x {f\left( t \right)dt}

    Here, f is still a function of x, but we use the dummy variable t to not mix integration variable and integration limit. The (-)F(a) plays the role of the constant of integration in the notation before, cfr. the fundamental theorem of calculus, of course.
    Now, since (*) may hold for more than one function (trivially: you can always add a constant), we indeed don't have "a primitive function", but a "set of primitive functions", which is exactly what we mean with the notation of the indefinite integral.
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    There is a problem here. There is a bounded function that is a well-defined derivative of another function. However, that function is not integrable using the Riemann or the Lebesque sense. Not until we get the Kurzweil/Henstock definition do we have an integral that has the anti-derivative property.
    Last edited by Plato; November 17th 2006 at 02:59 PM.
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    Quote Originally Posted by TD! View Post
    Given a function f(x) ([a,b] to R), we call a differentiable function F(x) ([a,b] to R) a primitive function of f(x), if F(x)' = f(x), for all x in [a,b].
    I think you meant (a,b) in both cases.
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    TD!
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    It's not a problem to define the function on a closed interval, but the differentiability is sufficient on the open interval.
    Let me rephrase if you want it a bit more formal.

    Given the functions f : [a,b] \to \mathbb{R} and F : [a,b] \to \mathbb{R}.
    The function F is called a primitive function of f, if the following holds:

    - F is continuous on [a,b]
    - F is differentiable on (a,b)
    - F'\left( x \right) = f\left( x \right), \; \forall x \in (a,b)
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