Formal definition of the integral?

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I've just realized that often, especially is physics, integrals are used merely as syntactic sugar representing an infinite sum of the form: lim(dx->0) sum(f(x)*dx) where dx is delta-x. Take for example the simplified equation for center of mass: cm = I(p dm)/tm where I is the integral symbol, p is the position of a point mass relative to the centroid of the body, dm is the mass of the individual point masses (and so p dm is the first moment), and tm is the total mass. The integral in that equation, as far I know, isn't really calculating an area, volume, etc. It's merely a fancy way of writing an infinite sum of the above form. What I'm wondering is how the integral is formally defined. I've looked it up on the internet and found it to always be described as the area under the curve, volume, curl, etc. Is it correct to say that, in a very broad sense, the integral is just fancy syntax for infinite sums of the above form?

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I'm not a math expert but this is the way I look at it.

The indefinite integral is the infinite sum: lim(dx->0) sum(f(x)*dx) = F(x).
The definite integeral is F(b)-F(a) which also "happens" to represents the area under f(x) from x=a to x=b.

I think it's important to not forget that the integral is an infinite sum. I've seen many students distribute roots inside the integral. If they would have known the definition they would have known that this is a no-no.

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Well, the integral is the anti-derivative. That formula is the Riemann sum.

I think a mathematician would most likely be annoyed by making very broad generalizations.

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Quote:
 Original post by nilknI've just realized that often, especially is physics, integrals are used merely as syntactic sugar representing an infinite sum of the form:lim(dx->0) sum(f(x)*dx)where dx is delta-x. Take for example the simplified equation for center of mass:cm = I(p dm)/tmwhere I is the integral symbol, p is the position of a point mass relative to the centroid of the body, dm is the mass of the individual point masses (and so p dm is the first moment), and tm is the total mass.The integral in that equation, as far I know, isn't really calculating an area, volume, etc. It's merely a fancy way of writing an infinite sum of the above form.What I'm wondering is how the integral is formally defined. I've looked it up on the internet and found it to always be described as the area under the curve, volume, curl, etc.Is it correct to say that, in a very broad sense, the integral is just fancy syntax for infinite sums of the above form?

Pretty much. What you are describing is called the Reimann integral. There are other definitions of integrals and that leads to measure theory. But integration is all about adding stuff up.

Perhaps try the definition in wikipedia

-Josh

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Quote:
 Original post by Name_UnknownI think a mathematician would most likely be annoyed by making very broad generalizations.

20th Century mathematics was mostly about generalization [smile]

-Josh

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Thanks for clearing that up.

jjd: Of all the many places I looked, wikipedia just wasn't among them! The definition provided there satisifed my curiosity. Thanks for the link.

Now that this great burden has been lifted I can continue with my life. [grin]

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Quote:
Original post by jjd
Quote:
 Original post by Name_UnknownI think a mathematician would most likely be annoyed by making very broad generalizations.

20th Century mathematics was mostly about generalization [smile]

-Josh

Ok, I think my Differential Equations professor would be very annoyed by broad generalizations ;-)

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Quote:
 Original post by Name_UnknownOk, I think my Differential Equations professor would be very annoyed by broad generalizations ;-)

Haha! [smile] I don't doubt... although I'm sure he finds the Sturm-Louiville theorem most satisfying [wink][wink]

-Josh

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Quote:
 Original post by nilknThanks for clearing that up.jjd: Of all the many places I looked, wikipedia just wasn't among them! The definition provided there satisifed my curiosity. Thanks for the link.Now that this great burden has been lifted I can continue with my life. [grin]

You're welcome! Calculus is actually a pet peeve of mine. I really don't like the way it is taught in most universities. One thing that I would love to see in a calculus course is historical material used to introduce and motivate the topics covered. The fundamental theorem of calculus is often presented after teaching techniques of integration and I think its a real anticlimax for the student. I mean, it's the FUNDAMENTAL THEOREM!!! But the fact that the area under a curve and the slope of the curve are so tightly related is truly weird! I mean, why would you expect this?!

OK, I'll hold my rant in... [wink]

-Josh

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I think it would be strange if the shape of the curve had nothing to do with the area under it.

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