Jump to content
  • Advertisement
Sign in to follow this  

Formal definition of the integral?

This topic is 4636 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Recommended Posts

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?

Share this post


Link to post
Share on other sites
Advertisement
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.

Share this post


Link to post
Share on other sites
Quote:
Original post by nilkn
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?


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

Share this post


Link to post
Share on other sites
Quote:
Original post by Name_Unknown
I think a mathematician would most likely be annoyed by making very broad generalizations.


20th Century mathematics was mostly about generalization [smile]


-Josh

Share this post


Link to post
Share on other sites
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]

Share this post


Link to post
Share on other sites
Quote:
Original post by jjd
Quote:
Original post by Name_Unknown
I 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 ;-)

Share this post


Link to post
Share on other sites
Quote:
Original post by Name_Unknown
Ok, 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

Share this post


Link to post
Share on other sites
Quote:
Original post by nilkn
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]


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

Share this post


Link to post
Share on other sites
Sign in to follow this  

  • Advertisement
×

Important Information

By using GameDev.net, you agree to our community Guidelines, Terms of Use, and Privacy Policy.

Participate in the game development conversation and more when you create an account on GameDev.net!

Sign me up!