Quick description
Sometimes one can calculate a definite integral such as (or more generally , where is a measurable set, is an integrable function, and is a measure) exactly by antidifferentiating and using the fundamental theorem of calculus. However, sometimes this method does not work, or is not appropriate. (The second might be the case if, for example, one was integrating not an explicit function but an unspecified function about which one had certain information.) Also, in many applications one does not actually need to compute an integral exactly; merely obtaining a sufficiently good approximation to that integral, or even just an upper bound on its magnitude, may be sufficient in many applications (particularly those in analysis).
This article gives links to further articles about techniques for approximating or bounding integrals. It is useful to make a distinction between nonoscillatory methods, which do not try to exploit cancellation in the sign or phase of the integrand , and oscillatory methods, which are designed to take advantage of such cancellation. For unsigned integrals, one should of course try the nonoscillatory methods; for signed integrals, the nonoscillatory methods tend to be easier but cruder (giving worse bounds), while the oscillatory methods give better bounds but are harder to implement.
Links to articles

General methods

Methods for bounding integrals (nonoscillatory)^{♦}

Methods for bounding integrals (oscillatory)
Comments
Inline comments
The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.
A quick question about this:
Mon, 20/04/2009  21:12 — gowersA quick question about this: is it your intention to chop up the existing "bounding integrals" article into a fairly large number of subarticles, as this suggests? I myself think that would be a good idea – each technique could then be illustrated by more than one example, and each article could be devoted to a single technique (which, though not an absolute requirement, is in general preferable I think).
Yes, I'm chopping it up
Mon, 20/04/2009  21:43 — taoThe article was getting rather unwieldy, and I was always relabeling the references between one method and another. It will lead to ten stubby articles rather than one overly long one, but this is probably better for the longer term development of these pages.
Excellent – I look
Mon, 20/04/2009  21:51 — gowersExcellent – I look forward to the results, and to making heavy use of links to these articles when I come to write the I have a problem about integration page.
From the point of view of
Thu, 23/04/2009  14:08 — gowersFrom the point of view of linking to this article, it is quite inconvenient to have levels of the hierarchy that are not articles, with sublevels that are. Would it be possible to create small articles for the headings above that are currently in black? (I could do this myself, but am rather busy for the next 24 hours or so.)