Talk:Integral: Difference between revisions

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imported>Catherine Woodgold
(→‎Why not a physical example?: summing small pieces of work may be a good example.)
imported>Aleksander Stos
(Arbitrary shapes)
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: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. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span> 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
: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. [[User:Robert Tito|Robert Tito]]&nbsp;|&nbsp;<span style="background:grey">&nbsp;<font color="yellow"><b>[[User talk:Robert Tito|Talk]]</b></font>&nbsp;</span> 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.  --[[User:Catherine Woodgold|Catherine Woodgold]] 21:38, 1 May 2007 (CDT)
::I think summing small pieces of work is a concept people might find it easy to grasp intuitively.  --[[User:Catherine Woodgold|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. --[[User:Aleksander Stos|Aleksander Stos]] 16:04, 3 May 2007 (CDT)

Revision as of 16:04, 3 May 2007

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)

Intuitively

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)