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First of all sorry for my English, I'm not used to communicate with this language. I want to ask something about a thing that I discovered while studying physics (AKA applied mathematics).

There is this strange operator $\int$ that integrates, this not so curly "S" that means "infinite sum of infinitesimal quantity/numbers", but not only it does "a sum", but it also takes advantage of the "infinitesimal environment" to "linearize" every kind of figure. Ok? (I don't even know if I'm explaining this in the right way...)

So this guy is kind of an analytic and a geometric operators mixed together, and here comes my question: What is the result of an integration?

Is the result of an integration still a mathematical law? Is it a parametric equation? What's the analytical difference between the two, omitting all the rest we can say about integrals and the entire math?

Usually I keep hearing that "the derivative lowers the grade and the integral raises", but this is not at all about the grade. Also, the grade is only part of an analytical dissertation about a law or a resolution of a problem.

I hope that I'm explaining my question in the right way, thanks for all the replies.

PS. I also think that if I understood this better, I could really understand what the integration constant means.

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    This is better asked at math, not physics, site. It is purely about the notion of integration. I would recommend Lang's calculus book, or perhaps a 19th century book on calculus (these are always easy and interesting historically, and give you a different perspective).2011-11-06
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    @ron-maimon can you be more specific? I'm really looking for a good answer that explain me this, and not one that just say this symbol stay for "bla bla bla" and does "bla bla bla". What is the title of the book you are suggesting?2011-11-06
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    What do you mean by "mathematical law"? Can you give any example?2011-11-06
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    @zev-chonoles i.e. law=log; function=(y=log(x))2011-11-06
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    @Micro: I'm sorry, but that answer only confuses me more than I was before.2011-11-06
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    A mathematical law can be an axiom, or a statement proven from the axioms. In Euclidean geometry one axiom is "the shortest distance between two points is a straight line"; Pythagoras's theorem can be proven from the axioms. Both of these are mathematical laws. A parametric equation on the other hand is the redefining of a variable in terms of another.2011-11-06
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    @Micro: Before going into concepts it may help to know if you have difficulties understanding the definition of the proper integral for integrand, i.e. defined and finite function on a closed bound interval?2011-11-06
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    @yauhen-yakimovich long story short: i have only an "aritmetic" notion of the integral operator, my teacher used to use a table for reference and i also do the math this way too, but i think this is a bad way to understand the math, especially when I'm try to learn physics like i'm doing now; i found this 2 disciplines relatively simple and easy but only when i figured it out what each notions really means, so i'm here asking for something i'm not able to properly understand. I can operate with integrals on a piece of paper, but i don't know what they means.2011-11-07

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