[Tex/LaTex] Lower and upper Riemann integrals

math-modemath-operatorssymbols

How can I make an integral symbol with a bar above it or a bar below it?

Best Answer

From what I've seen online, it suffices to use \overline and \underline.

Here is a minimal example that defines

\upRiemannint{<lo>}{<hi>}

which draws the "upper Riemann integral" over the range [<lo>,<hi>]. Analogously,

\loRiemannint{<lo>}{<hi>}

defines the "lower Riemann integral" over the range [<lo>,<hi>].

enter image description here

\documentclass{article}
\newcommand{\upRiemannint}[2]{
  \overline{\int_{#1}^{#2}}
}
\newcommand{\loRiemannint}[2]{
  \underline{\int_{#1}^{#2}}
}
\begin{document}
\[
  \loRiemannint{a}{b} f(x)\,\mathrm{d}x \qquad \textrm{or} \qquad \upRiemannint{a}{b} f(x)\,\mathrm{d}x
\]
\end{document}

These integrals also translate to use in text mode, but vertical alignment is slightly off due to the integral sign by default.

Related Solutions

[Tex/LaTex] XOR and II (concatenation) summation symbols

You want to use \bigoplus instead of \oplus.

I've never seen concatenation done that way. Addition and XOR are commutative operations so it makes sense to sum over a set (or take the exclusive OR of a set). Concatenation is not like that. I think I would explicitly write out the concatenation. That said, you can use \bigparallel from the stmaryrd package.

\documentclass{article}
\usepackage{amsmath}
\usepackage{stmaryrd}
\newcommand*\concat{\mathbin{\|}}
\begin{document}
\[x_1\concat x_2\concat\dotsb\concat x_n\]
\[\bigparallel_{i=1}^n x_i\]
\end{document}

enter image description here

[Tex/LaTex] Writing the Existential Quantifier with Lower and Upper Limits

Here is a solution which is similar in spirit to Mico's earlier answer, condensed into a few lines of code. This requires amsmath.sty and scalefnt.sty. (Thanks to egreg for remarking on multiple improvements to the original answer; I've condensed it further based on his remarks.)

\newcommand\ScaleExists[1]{\vcenter{\hbox{\scalefont{#1}$\exists$}}}

\DeclareMathOperator*\bigexists{%
  \vphantom\sum
  \mathchoice{\ScaleExists{2}}{\ScaleExists{1.4}}{\ScaleExists{1}}{\ScaleExists{0.75}}}

The choices of font size were achieved by experimentation, and should be adequate for normalsized Computer Modern at 10pt to 12pt. Minor tweaking may be necessary for other typefaces. Because other font sizes (such as \Large and \footnotesize) are achieved by scaling up the size of normal characters, this should also work in other font sizes up to the limits of \scalefont to change the character size (up to about \LARGE at 11pt).

The parameters were chosen to achieve as similar an appearance as possible to the size and alignment of \sum in each context (displaystyle, textstyle, scriptstyle, and scriptscritstyle), with the same default behaviour of \limits and \nolimits in each context. In particular, the \vphantom\sum at the beginning of \bigexists is used to achieve precisely the same vertical spacing of the limits from the \exists symbol, as the limits would otherwise be closer to the symbol than they should be. Here is how it looks at 10pt:

$Example presentation of <code>\bigexists</code> macro$

This same code should be similarly generalized to any symbol of a similar height and depth (e.g. \forall).

It should be easy to tweak the results to obtain better tuning for other fonts or point sizes; additional refinements (or defining a custom character with \METAFONT) may be necessary to obtain a more robust solution.

Best Answer

Related Solutions

[Tex/LaTex] XOR and II (concatenation) summation symbols

[Tex/LaTex] Writing the Existential Quantifier with Lower and Upper Limits

Related Question