# Lower and upper Riemann integrals?

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

• 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

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>].

\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.

• 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
• @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

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}


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.

(from unimath-symbols doc)

• Do we need \mathchoice for \lowint too? – Tuff Contender Apr 8 '18 at 17:42

If you want complete control over the placement of the bars, you can try the following code for the lower integral:

\lefteqn{\int_a^b f(x)}\lefteqn{\hspace{0.0ex}\rule[-2.25ex]{1.1ex}{.05ex}}
\phantom{\int_a^b f(x)}\mathrm{d}x


And for the upper integral:

\lefteqn{\int_a^b f(x)}\lefteqn{\hspace{1.2ex}\rule[ 3.35ex]{1.1ex}{.05ex}}
\phantom{\int_a^b f(x)}\mathrm{d}x


The hspace argument controls left/right positioning. The three rule arguments respectively control height, length and thickness of the bar.

• What package (or otherwise) provides \lefteqn? – Werner Feb 20 '17 at 17:11
• @Werner It's in the kernel. – egreg Feb 20 '17 at 17:13

Another non UTF option is to use \upint and \lowint commands available in stix-package. Read section 2.2 of the documentation that comes with it, particularly compatibility with AMS packages.