Two Examples of Improper Integrals

In a recent posting I proposed to answer a question using an indefinite integral, which would more-correctly have been solved using the corresponding, definite integral. The issue there was that if this integral was rewritten as some arbitrary definite integral, this could in some cases have resulted in what’s called an ‘improper integral’. And what my reader may not realize, is that improper integrals exist, with well-behaved solutions, just as some infinite series converge.

And so, I have written a work-sheet below, which reminds people who may not remember their Calculus 2 exactly, of what forms improper integrals can take:

Link to a Letter-Sized PDF File

Link to an EPUB File for Phones

(Edit 6/05/2019, 18h25 : )

I have just revised the work-sheets above, to include some plots, and to provide a clearer understanding to anybody who might be interested in them, but who did not study Calculus 2. But some readers of the EPUB version may notice wonky formatting.

When I export Math notation to regular HTML, or to anything which is based on regular HTML, such as to an EPUB File which is not using MathML, I am faced with a problem every time correct Mathematical notation requires that 3 glyphs be stacked, as is the case with the (definite) integral operator, and with the Sigma operator, the latter of which denotes a summation. The only way I see around this issue is, to give the operator in question both a subscript and a super-script.

While the result can be read and understood, doing so requires additional concentration by the reader. I’ve written earlier postings, in which I described this problem, but the advantages here are, a notation which regular EPUB readers can display, as well as my ability to include the Computer Algebra and thus the plots, of “SageMath”, using the “LyX” graphical front-end to LaTeX, which makes the typesetting easier for me.

This limitation does not exist when only exporting the results to a PDF-File. But, in order to take advantage of the more-correct formatting of the resulting PDF File, I’d need to create two separate versions of my own document, one for export to (exact) PDF, and one for export to (messy) EPUB. While I did this amount of work for simpler work-sheets, I’m unwilling to do this for the more-complex work-sheet I just linked to.

(Edit 6/05/2019, 18h55 : )

There’s an added challenge to me, in the form of something my particular software is unable to perform. When I’m using LyX to typeset my work-sheets, the following two possibilities emerge:

• Those work-sheets may consist entirely of Math written in my own hand, in which case I am able to export them to an XHTML File that contains MathML, and this will enable me to set up a master document, in which 2 but not 3 elements can be stacked. Then, the ‘Limus’ notation will be correct, and the integrals will be so-so. But the resulting master document can then be exported in two ways that eventually end up as a PDF and as an EPUB3 File, the latter requiring MathML from the EPUB reader app.
• Those work-sheets may contain Computer Algebra and/or Plots that are essential, in which case only .TEX Files of the SageTex variety can be exported, which in turn can only be converted into plain HTML. This will result in an EPUB File that is inferior, but that all mobile EPUB reader apps can view. But simultaneously, through a separate master document and additional work on my part, a pristine PDF File can result, which still requires a full-sized monitor or other output device to read.

So, unless I find ways to export SageTex Files specifically, to XHTML with MathML, I’ll be facing issues in how to create typeset documents in the near future.

Dirk