Definition talk:Definite Integral

The geometric interpretation part of this is exactly what I had in mind for the geometric Riemann sum. But it seems silly to just copy and paste. The concepts are related but not identical. Any ideas?--GFauxPas 17:49, 18 January 2012 (EST)
 * Nevermind, I have an idea. --GFauxPas 19:58, 18 January 2012 (EST)
 * I think the approach would be to include the Riemann Sum as a transcluded section in the Definite Integral page so as to maximise the use of common material. --prime mover 01:52, 19 January 2012 (EST)
 * Well let me say my idea and you'll tell me what you think. The upper and lower sums are examples of Riemann Sums, but I can make a picture where the choice of $(c_i, f(c_i))$ is completely arbitrary, i.e. somewhere in the middle of the sub interval. Perhaps I can even put a discontinuity on the graph, just to emphasize that the Riemann Sum doesn't require continuity. --GFauxPas 08:08, 19 January 2012 (EST)

Riemann integral?
This article only deals with the Riemann integral (not other types of integration such as the Lebesgue integral). As such, should the page name be "Riemann integral" instead of "definite integral"? Also, I changed the definition presented in the page because the previous definition was incorrect, if I'm not mistaken somewhere. Could somebody check whether this (new) definition is correct, and if at least some of the material is still relevant to the source mentioned (I don't have the book)? Abcxyz 22:37, 18 March 2012 (EDT)
 * a) We're back to "What would an elementary-level student understand?" The introductory level integration as taught in schools is precisely what is on this page. Later on, when measure theory is encountered (postgrad level, I understand) then different models of integration are encountered. The simple and straightforward model based on the concept of piecewise continuity is then relabelled "Riemann integral" while the more complicated stuff is called a "Lebesgue integral" or some such. (I'm going to be told I'm wrong here, I gave up caring for lent so bring it on). So renaming this page "Riemann Integral" is going to lose us the users at undergraduate level and below - possibly our biggest domain. We have already invoked Riemann inside the page, and there's a redirect from Definition:Riemann Integrable, that should be enough.
 * b) Exactly how was the original page "incorrect"? As the argument came from the published source, we need to add a page to the "Mistakes in Published Works" category and explain exactly what the problem is. If the wrongness is merely that it glosses over a detail, then I recommend we add that detail (if necessary as a link), rather than just throw away what's already there and replace it with something else which is not backed up by a source work. --prime mover 03:13, 19 March 2012 (EDT)


 * I did not mean to put Lebesgue integration in the same article. If you think that the current way is the best, then by all means keep it so. I appreciate your input. Again, I apologize for any confusion I could have caused. Abcxyz 10:00, 19 March 2012 (EDT)


 * Goodness, certainly not. That was not what I was suggesting. We already have a page for Lebesgue integral.
 * My suggestion as to the way to go was effectively: leave it (mainly) the way it is, with perhaps a little more wordage on the fact that it's called a "Riemann Integral" with perhaps a redirect from a page with that name. We can add an also-see to Lebesgue integral. My point was that renaming this page to "Riemann Integral" would not be what we want to do, because this concept is best known as "definite integral", assuming that those of B.Sc level and below outnumber postgrads.
 * As for the justification for the original definition, once the issue as pointed out by Lord_Farin has been resolved, we should be able to have both definitions up. I'll attend to it later.
 * Again, the message continues: it is better to add an alternative way of doing a job (definition or proof, whatever) than replacing what's there with something different - particularly if there is a citation to a reference in published literature. --prime mover 11:13, 19 March 2012 (EDT)


 * Got it. Abcxyz 11:35, 19 March 2012 (EDT)

The problem appears to be that when the Riemann integral exists, it equals the $y$ in the previous definition; however, if the Riemann integral doesn't exist (standard example: the indicator function of $\Q$ on $[0..1]$) then the upper and lower sums may genuinely stay apart, making $y$ ill-defined in these cases. However, I particularly liked that formulation and hopefully a small formal fix will do the trick. I was thinking of saying 'the unique $y$ [...]' and then adding a statement that if multiple, distinct $y$ satisfy the definition, that then the integral isn't defined. The definition given by abcxyz also works, but I'd rather see them proven equivalent (as I have relied on the previous formulation in some proofs already). --Lord_Farin 03:46, 19 March 2012 (EDT)
 * So it's worth retaining the original definition then, with that specific fix, and adding the new one as a separate definition (separate page, transcluded of course). --prime mover 05:44, 19 March 2012 (EDT)