Lower and upper Riemann integrals?

Question

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

Answer 1

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.

Answer 2

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)

Answer 3

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.

Answer 4

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.