# 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 '11 at 17:07
• I came here interested in typesetting, and found it useful. – Ross Millikan Jun 15 '11 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'. – night owl Apr 15 '11 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. – Danie Els Apr 15 '11 at 15:35
• Thank You. That was very useful to know for later references. – night owl Apr 15 '11 at 15:56
• @night: \tfrac is nice for numerical fractions but I would avoid it when variables are in the numerator or denominator. – Matthew Leingang Apr 15 '11 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 $$\displaystyle\int f(x)\ dx.$$ – night owl Apr 19 '11 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 – night owl Apr 15 '11 at 15:57
• You could just write \left.\frac{1}{2} x^2\right|_a^b, too. – Ryan Reich Apr 15 '11 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 '11 at 15:32

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. – Yiannis Lazarides Apr 16 '11 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. – romeovs Apr 16 '11 at 18:15
• This is the standard way of typesetting integration limits in Russian literature. – ScumCoder Aug 16 '16 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. – Chappers May 27 '17 at 21:48

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} 