Talk:Integral

Formal definition of the Riemann integral

This is a small point. I spotted a mistake in the definition of the Riemann integral, which included the following segment:

For all $\varepsilon >0$ there exists $\Delta _{i}>0$ such that, for any tagged partition $[a,b]$ with mesh less than $\Delta _{i}$,

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .$

This is a typical argument of the epsilon-delta type. The mesh of a partition is the width of the largest sub-interval formed by the partition. If the width of the largest sub-interval (with some index k which we don't need to know) is $<\delta$, this implies that for all sub-intervals $\Delta _{i}$ are $<\delta$. No need to go at the level of indices or of taking into account the plurality in the notation: the notion of mesh does the job.

So the correct formulation should be (and using lower case delta makes the argument even clearer, showing that it is the familiar epsilon-delta argument):

For all $\varepsilon >0$ there exists $\delta >0$ such that, for any tagged partition $[a,b]$ with mesh less than $\delta$,

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .$

I will try again, asking all those who want to revert my change to read the above comment and indicate where it goes wrong, if you find something wrong with it.

Dessources (talk) 13:13, 15 August 2021 (UTC)

How can something like "There exists delta such that <expression involving Delta_i>" be possible correct? I have reverted again. - DVdm (talk) 14:05, 15 August 2021 (UTC)

Could you please be more precise in the formulation of your objection. What expression are you referring to?

If you mean the following expression:

$\left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon$ (*)

please note that this expression does not contain $i$ as a free variable. Here $i$ is an index that runs from 1 to $n$, and expression

$\left|S-\sum _{j=k}^{n}f(t_{k})\,\Delta _{k}\right|<\varepsilon$

is strictly equivalent to it.

Dessources (talk) 14:59, 15 August 2021 (UTC)

I see what you mean. The $\Delta _{i}$ are defined in the previous paragraph. Let me propose a version that may accommodate your concern before reverting my change.Dessources (talk) 15:07, 15 August 2021 (UTC)

I did not find a way to make things clearer without repeating what is already said. When we say (*) is satisfied for any tagged partition, we refer to the definition just given of a tagged partition. Any tagged partition is defined by the finite increasing sequence of $x_{i}$'s, from $a$ to $b$, described in the first paragraph, which mechanically imply the width of the sub-intervals, Δ_i = x_i−x_i−1, and the $t_{i}$'s that fall within these intervals. When we take any tagged partition, we automatically get the $x_{i}$'s and $t_{i}$'s that define it, and thus also the $\Delta _{i}$'s.

Finally the error I corrected is obvious when one observes that it makes no sense to refer outside an expression to a variable which is bound in the expression, as was the case with the index $i$ that I removed. This alone is sufficient to justify the correction.

Dessources (talk) 15:52, 15 August 2021 (UTC)

@Dessources: Ah yes, I fumbled with the verification of the cited source. I overlooked the actual definition in section 8.5. I only had a look at the top of the page. My bad! - DVdm (talk) 07:56, 16 August 2021 (UTC)

Analytical vs Symbolic?

[1] breaks out separate sections for analytical vs symbolic integration, but I was raised that analytical and symbolic mean the same thing in this context. Is there some different meaning I'm not aware of? Rolf H Nelson (talk) 04:56, 4 January 2022 (UTC)

I agree. It's not clear what the distinction is supposed to be. In fact, there is a great deal of overlap in the content, as it is currently written. Unless somebody chimes in with a strong explanation, I'd support merging the two sections. Mgnbar (talk) 13:40, 4 January 2022 (UTC)

I also agree that the article is muddled: finding an antiderivative is described in both the "Analytical" and "Symbolic" subsections. The article can be improved.

• The main division is usually between those methods that find a formula containing well-known functions, and those methods that directly find a numerical value. I have seen the latter methods referred to as "approximate integration". But Wikipedia already has articles on symbolic integration and numerical integration so they are probably the best terms to use.

• I have seen methods of solving differential equations classified as "graphical", "numerical" or "analytical". But I am not sure how much the term analytical integration is used. There could be a distinction between methods that use clever mathematical analysis thinking and those that use brute-force calculation. Or perhaps analytical integration only applies to analytic functions. The term symbolic integration is probably becoming more popular because it is used in computer algebra systems.

• The current "Analytical" and "Symbolic" subsections both mention methods that find a symbolic representation as an infinite series which is then evaluated numerically. I have seen such methods classified as "approximate", and they could go in the "Numerical" subsection. But it might be better to have them in a separate subsection under the traditional name "Integration by series".

• I do not agree with the "Analytical" subsection where it says that "The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus". A numerical method such as counting squares under a graph is much simpler to explain.

• Also, I would change the name of the "Mechanical" subsection to "Analogue" and include links to both differential analyser and electronic analogue computer. JonH (talk) 16:43, 16 May 2022 (UTC)

Integration - calculas

Finding area by integration on the area between curve y = f(x) and x-axis? — Preceding unsigned comment added by 129.232.97.252 (talk) 16:56, 14 May 2022 (UTC)

Proposed additions, sections shown, thanks

Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.

The bracket integration method is a generalization of Ramanujan's master theorem that can be applied to a wide range of integrals.

• Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020), "An extension of the method of brackets. Part 2", Open Mathematics, 18 (1): 983–995, doi:10.1515/math-2020-0062, ISSN 2391-5455

• Rich, Albert; Scheibe, Patrick; Abbasi, Nasser (16 December 2018), "Rule-based integration: An extensive system of symbolic integration rules", Journal of Open Source Software, 3 (32): 1073, doi:10.21105/joss.01073

TMM53 (talk) 08:23, 2 January 2023 (UTC) TMM53 (talk) 08:23, 2 January 2023 (UTC)

I added this content and 2 references.TMM53 (talk) 03:12, 23 March 2023 (UTC)

Lead Sentence

I am trying to improve the lead sentence since it came up in the village pump as an example of something that needs work. My contribution is based on the suggestions from a WikiProject:Mathematics discussion Thenub314 (talk) 16:19, 10 February 2023 (UTC)