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

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Mathematical FYI: Technically they are lower and upper Darboux integrals. The Riemann integral is defined using tagged partitions instead. Of course the two definitions are equivalent as per the sketch of a proof given in the first Wikipedia article. –  kahen Feb 12 '12 at 6:08
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2 Answers

up vote 11 down vote accepted

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.

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Sorry for this negative comment but the output of your code just doesn't look right to my mathematical eye. Leo Liu got the right output. I wonder if the his code could be simplified? –  Predrag Punosevac Feb 13 '12 at 1:54
1  
@PredragPunosevac: I agree. I found this source and went from there... –  Werner Feb 13 '12 at 15:22
    
@PredragPunosevac yes, it may look bad, but that's how it is expressed in lots of books. –  Alfredo Hernández Mar 7 '13 at 16:31
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Personally I use shorter bars. It makes the macro much more complex:

\documentclass{article}
\usepackage{amsmath}
\def\upint{\mathchoice%
    {\mkern13mu\overline{\vphantom{\intop}\mkern7mu}\mkern-20mu}%
    {\mkern7mu\overline{\vphantom{\intop}\mkern7mu}\mkern-14mu}%
    {\mkern7mu\overline{\vphantom{\intop}\mkern7mu}\mkern-14mu}%
    {\mkern7mu\overline{\vphantom{\intop}\mkern7mu}\mkern-14mu}%
  \int}
\def\lowint{\mkern3mu\underline{\vphantom{\intop}\mkern7mu}\mkern-10mu\int}
\begin{document}
\begin{gather*}
   \upint_a^b f(x)\,\mathrm{d}x \\
   \lowint_a^b f(x)\,\mathrm{d}x
\end{gather*}
\end{document}

<code>\upint</code> and <code>\lowint</code>


I'm not sure which is better. Some new Unicode math fonts (XITS Math and Asana Math) have \lowint and \upint, and you can use unicode-math package to load the fonts. They also use wide bars.

enter image description here (from unimath-symbols doc)

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