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 Definition A central concept in calculus that generalizes the idea of a sum to cover quantities which may be continuously varying. [d] [e]
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 Workgroup category Mathematics [Categories OK]
 Talk Archive none  English language variant Not specified

Totality vs size

"Totality" might be better because integrals also describe such concepts as mass. But it's really hard to come up with a formulation that is both easy to grasp and accurate. Fredrik Johansson 13:54, 29 April 2007 (CDT)

I agree. "size" is not necessarily the best. Change it back to "totality" if you like. There may be something better. "Extent in space" doesn't cover all cases, either: one might want to integrate prices or interest rates or temperatures or something else, but since it says "intuitively" I think "extent in space" is good enough for that part -- it helps the reader get an image in their mind. I'll try to think of other words. --Catherine Woodgold 14:03, 29 April 2007 (CDT)
"Intuitively, we can think of an integral as a measure of the totality of an object with an extent in space. "
"... as a measure of the totality of some aspect, such as area or volume, of an object with an extent in space."
"... as a measure of some additive quality of an object."
"... as a measure of qualities such as area or volume, of the type whose values add when two objects are joined into a larger object."
"... as a measure of such qualities as area and volume."
"... as a way of extending the definition and measurement of area and volume to curved objects."
OK, I give up: leave it as "totality". I changed it back to the original. --Catherine Woodgold 18:35, 29 April 2007 (CDT)

Maybe you should just note that integrals generalize sums to (possibly) continuously varying quantities. Greg Woodhouse 13:47, 30 April 2007 (CDT)

The first sentence could be "An integral generalizes the idea of a sum to cover quantities which may be continuously varying, allowing for example the area or volume of curved objects to be calculated." --Catherine Woodgold 18:51, 30 April 2007 (CDT)


Please can somebody explain to me why I would Intuitively see integral as the way described in the first line ot the article? In my (and yes I am playing advocate of the devil) notion integral means total/aggregated. Can we put it into simpler lingo? Robert Tito |  Talk  19:34, 30 April 2007 (CDT)

Feel free to edit. Fredrik Johansson 12:06, 1 May 2007 (CDT)

Why not a physical example?

The opening paragraph mentions work (I think). Why not work out a simple example, like the work involved in drawing a bow string compared with the energy imparted to the arrow when the bow is released? How much fuel does it take for a rocket to reach the moon, bearing in mind that it is burning off fuel the whole time? Greg Woodhouse 18:39, 1 May 2007 (CDT)

seems a good idea, it might become more easier to see the change from summing small pieces of work to the total work needed or done on a system. Making the change from ∑ to ∫ more clear in the step from descriptive to analytical. Robert Tito |  Talk  19:29, 1 May 2007 (CDT) It can even be something simple as the work done to move a rock of 1 kg from 0 elevation to 10m elevation. Seems simple and intuitive to do? your thoughts? Rob
I think summing small pieces of work is a concept people might find it easy to grasp intuitively. --Catherine Woodgold 21:38, 1 May 2007 (CDT)

Arbitrary shapes

I find the prose of the article really compelling. Just hard to stop to read.

One remark, though. The sentence "[integration] allows us to exactly calculate lengths, areas, volumes — and so on, of arbitrarily complicated shapes" seems oversimplified (well, not true as it stands). I mean e.g. non-rectifiable curves (of infinite length) or non-integrable functions. So I propose changing to e.g. quite complicated shapes (maybe someone could find a better formulation). I think non integrable functions deserve to be explicitly mentioned (maybe an example of e.g. Dirichlet function that assigns 1 to rationals and 0 otherwise?). BTW, this gives a natural explanation why the Riemann integral is not the only one. IMHO, existence of other definitions of integral (e.g. Lebesgue) also should be mentioned too. --Aleksander Stos 16:04, 3 May 2007 (CDT)

Thanks! I had thought of mentioning non-integrable functions under "Technical definitions". Fredrik Johansson 17:58, 3 May 2007 (CDT)
I agree with Aleksander Stos. I think Fredrik Johansson is doing a great job writing this article. I've changed "arbitrarily" to "quite" as suggested, hoping Fredrik Johansson doesn't mind. I think there are cases where we can write a quantity as an exact formula but we don't know how to write its integral as an exact formula, so "arbitrarily" is a bit of an exaggeration. Even as it stands it could be taken to imply that we can calculate integrals for every case ("provided of course that") but I think it's OK like that anyway. (Oh-oh -- is "OK" an acronym?  :-)
Re this sentence (1st paragraph): For example, integrals can be used to calculate the length, area or volume of curved objects. I included "line" here and am considering possibly taking it out again. The idea is of measuring the length of a curved line, but in this sentence it's obvious to the reader, I think, what the problem is in finding the area or volume of a curved object, but the length of a curved object could be taken to mean something like the diameter of a circle: easy to measure and not what is meant. So maybe just deleting "line" here. Or maybe leaving it in -- I'm not sure.
Re this next bit: An integral might also measure one quantity that depends, in a cumulative way, on another quantity that is varying: ... I would change "might also measure" to "measures". --Catherine Woodgold 20:01, 3 May 2007 (CDT)