# Evaluation of Differentiation and Integration

I was having some thought about what would be the syntax to make the end point evaluation of derivatives or integrals. Such as making the | with the two end points of evaluation on the top an bottom of the line. Any suggestions would be wonderful.

• I was really hoping this was going to be a question about doing symbolic integration and differentiation in TeX. Alas, it was actually about typesetting. =)
– TH.
Apr 15, 2011 at 17:07
• I came here interested in typesetting, and found it useful. Jun 15, 2011 at 5:07

The \big| or \Big| symbols work quite well

\documentclass{article}
\usepackage{amsmath}
\begin{document}
$\int_a^b x^2\;\mathrm{d}x= \tfrac{1}{3} x^3 \Big|_a^b$
\end{document} • Thank You Danie, that worked excellent. Just what I was looking for. One question while on the topic. I noticed you using \tfrac, is there any significance to that over using \dfrac. How different are they and what do they actually mean if you know? Meaning the 't' and 'd'. Apr 15, 2011 at 15:13
• @night owl: \tfrac is the inline "text" mode fraction and \dfrac the display math frac. \tfrac is smaller and I prefer it for single line equations. Apr 15, 2011 at 15:35
• Thank You. That was very useful to know for later references. Apr 15, 2011 at 15:56
• @night: \tfrac is nice for numerical fractions but I would avoid it when variables are in the numerator or denominator. Apr 15, 2011 at 18:48
• How do you get that integral sign like that. Mines looks loose and sloppy at the ends. Using: $\displaystyle\int f(x)\ dx.$ OR $$\displaystyle\int f(x)\ dx.$$ Apr 19, 2011 at 11:54

The \left - \right construct gives you an expandable evaluation symbol:

\documentclass[12pt]{article}
\usepackage{amsmath}

\DeclareMathOperator{\di}{d\!}
\newcommand*\Eval{\left.#1\right\rvert_{#2}^{#3}}

\begin{document}

$\int_{a}^{b}x\di x = \Eval{\dfrac{1}{2}x^{2}}{a}{b}$

$\int_{a}^{b}\di x = \Eval{x}{a}{b}$

\end{document} EDIT: I modified the code following the comment by Ryan Reich.

• Thanks for the response. Your looks clean, just a tad bit more work. hehe.. +1 Apr 15, 2011 at 15:57
• You could just write \left.\frac{1}{2} x^2\right|_a^b, too. Apr 15, 2011 at 18:13
• +1 I prefer this over using the \big| or \Big| symbols because of the expandability - you may have a very complex expression of which you don't know the vertical extent, and hence which size modifier for the |. It's a "tad bit more work" up front as @night owl pointed out, but you don't need to tweak and recompile to get it looking just right :).
– cm2
Sep 27, 2011 at 15:32
• @RyanReich Why the dot is needed after \left? May 8 at 12:20
• @anton it's a placeholder because there has to be an argument there, but we don't actually want anything printed. It is a special case specifically designed for this purpose; no period is actually shown. May 8 at 19:28

If you want the evaluation symbol to be of the same height as the integration symbol, you can enclose a phantom integration symbol between \left and \right, like this:

\documentclass{article}
\usepackage{amsmath}

\DeclareMathOperator*{\di}{\mathrm{d}\!}
\def\at{
\left.
\vphantom{\int}
\right|
}

\begin{document}
\begin{eqnarray}
\int_a^b x^2 \di x &=& \frac{x^3}{3}\at_a^b \\
\int\limits_a^b x^2 \di x &=& \frac{x^3}{3}\at_a^b
\end{eqnarray}
\end{document} The commath package has the \eval command for this purpose.

This hasn't got anything to do with your question, but I'll post it anyway (it's on topic I guess).

I don't know if this is correct (since few people do it), but I like it when the integration limits are above and below the integral sign:

\DeclareMathOperator{\di}{d\!}

$\int\limits_a^b\! x\di x = \tfrac{1}{2}x^2\Big|_a^b$


As you can see, adding the macro \limits to your code makes the integral look good. You can do this for any math operator.

Also notice that the \! command brings the integrandum closer to the integral sign. I like this kind of snugged integrals. • .. unusual, is there a maths book or article using it? I would be interested in having a look. Apr 16, 2011 at 17:22
• @Yiannis: As I said, don't know if it's the correct way of typesetting integral bounds, I just like it myself. I geuss I've seen it around somewhere, just can't remember where. Apr 16, 2011 at 18:15
• This is the standard way of typesetting integration limits in Russian literature. Aug 16, 2016 at 21:26
• This is probably not done because the integral symbol's already pretty tall, so stacking over- and underscripts on it fills even more vertical space. May 27, 2017 at 21:48

None of the stuff here gave me the desired outcome in the editor I was using (MATLAB live script), but I found this and it's really nice:

\bigg/_{\!\!\! a}^{\,b}


Example:

\frac{\sqrt{\pi}}{2}\bigg/_{\!\!\! a}^{\,b}x^2 • It is not ISO compliant. Mar 5, 2021 at 9:14
• I don't know what that means in this context, but maybe someone finds that useful to know. Mar 5, 2021 at 15:46