\int with \limits or without?

Question

I read that I can use \int_a^b or \int\limits_a^b. With \limits, the equation looks more compact horizontally, but a little bigger vertically.

Is there some design rule to it, or just personal preference?

Answer 1

I am not aware of explicit design rules that would stipulate when to use side-set and when to used below/above limits of integration. That said, it's very rare to see anything but side-set limits of integration in either inline or display math mode.

The main reason for not placing the limits of integration above and below the integral sign must surely be that doing otherwise would increase considerably the depth and height of the expressions, which greatly risks ruining the overall "color" (more precisely: the average grayness) of the page on which the expressions are typeset.

The only exception to this dictum I can think of -- at least for single integrals -- is if the integrand itself is quite large, e.g., if it contains a double-fraction term. In such cases, placing the limits of integration above and below the integral symbol could help simplify the visual experience of the entire expression. As @PeterGrill and the other answer highlight, two further good candidate cases for setting the limit of integration below the integral symbol(s) are (i) if one is dealing with multiple integrals and (ii) if one wishes to express the entire set over which the integration takes place with a symbol (e.g., \mathbb{R}) rather than with explicit lower and upper bounds.

The following example, in which all expressions are typeset in display math mode, contrasts the visual appeal of three separate integral expressions: the Gamma function, the Beta function (in a form that involves a fractional term), and an entirely fictitious integral expression that involves a double fraction term. Speaking for myself, I'd stay that only in the third case is it defensible to use \int\limits instead of just \int. Observe that the example code also demonstrates that issues of excessive amounts of white space between the integral symbol (and its side-set limits of integration) and the integrand are best dealt with by applying one or more \! (negative thinspace) instructions rather than by setting the limits of integration above and below the integral symbol.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\Gamma(z)=\int_0^\infty \!\! e^{-x}x^{z-1}\,dx 
  \quad&\text{vs.}\quad 
  \Gamma(z)=\int\limits_0^\infty e^{-x}x^{z-1}\,dx\\
B(x,y) = \int_0^\infty \!\! \frac{t^{x-1}}{(1+t)^{x+y}}\,dt
  \quad&\text{vs.}\quad
  B(x,y) = \int\limits _0^\infty \frac{t^{x-1}}{(1+t)^{x+y}}\,dt\\
\int_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\,dx 
  \quad&\text{vs.}\quad
  \int\limits_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\,dx
\end{align*}
\end{document}

Answer 2

I have never seen \int\limits used in syllabi at my university unless it was used for double (\iint) or more combined integrals. In the code fragment below, without the \limits, it is formatted wrong.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\iint\limits_D \quad \iint_D
\end{equation}
\end{document}