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)
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)