> For the complete documentation index, see [llms.txt](https://overleaf-pro.ayaka.space/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://overleaf-pro.ayaka.space/latex/mathematics/09-integrals-sums-and-limits.md).
# Integrals, sums and limits
## Integrals
Integral expression can be added using the `\int_{lower}^{upper}` command.
Note, that integral expression may seems a little different in inline and display math mode.
| LaTeX code | Output |
| ------------------------------------------------ | ------------------------------------------------------------------ |
| `Integral \(\int_{a}^{b} x^2 \,dx\) inside text` |  |
| `\[ \int_{a}^{b} x^2 \,dx \]` |  |
## Multiple integrals
To obtain double/triple/multiple integrals and cyclic integrals you must use `amsmath` and `esint` (for cyclic integrals) packages.
| LaTeX code | Output |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------- |
| `latex
\begin{gather*}
\iint_V \mu(u,v) \,du\,dv
\\
\iiint_V \mu(u,v,w) \,du\,dv\,dw
\\
\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\\
\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\end{gather*}
` |  |
| `latex
\[
\oint_V f(s) \,ds
\]
` |  |
## Sums and products
Like integral, sum expression can be added using the `\sum_{lower}^{upper}` command.
| LaTeX code | Output |
| -------------------------------------------------- | --------------------------------------------------------------------- |
| `Sum $\sum_{n=1}^{\infty} 2^{-n} = 1$ inside text` |  |
| `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` |  |
In similar way you can obtain expression with product of a sequence of factors using the `\prod_{lower}^{upper}` command.
| LaTeX code | Output |
| -------------------------------------------- | ----------------------------------------------------------------------- |
| `Product $\prod_{i=a}^{b} f(i)$ inside text` |  |
| `\[ \prod_{i=a}^{b} f(i) \]` |  |
## Limits
Limit expression can be added using the `\lim_{lower}` command.
| LaTeX code | Output |
| -------------------------------------------- | ------------------------------------------------------------- |
| `Limit $\lim_{x\to\infty} f(x)$ inside text` |  |
| `\[ \lim_{x\to\infty} f(x) \]` |  |
## Examples
Here is an example LaTeX document:
```latex
\documentclass{article}
\title{Integrals, Sums and Limits}
\author{Overleaf}
\date{}
\usepackage{amsmath}
\begin{document}
\maketitle
\section{Integrals}
Integral \(\int_{a}^{b} x^2 dx\) inside text.
\medskip
The same integral on display:
\[
\int_{a}^{b} x^2 \,dx
\]
and multiple integrals:
\begin{gather*}
\iint_V \mu(u,v) \,du\,dv
\\
\iiint_V \mu(u,v,w) \,du\,dv\,dw
\\
\iiiint_V \mu(t,u,v,w) \,dt\,du\,dv\,dw
\\
\idotsint_V \mu(u_1,\dots,u_k) \,du_1 \dots du_k
\\
\oint_V f(s) \,ds
\end{gather*}
\section{Sums and products}
Sum \(\sum_{n=1}^{\infty} 2^{-n} = 1\) inside text.
The same sum on display:
\[
\sum_{n=1}^{\infty} 2^{-n} = 1
\]
Product \(\prod_{i=a}^{b} f(i)\) inside text.
The same product on display:
\[
\prod_{i=a}^{b} f(i)
\]
\section{Limits}
Limit \(\lim_{x\to\infty} f(x)\) inside text.
The same limit on display:
\[
\lim_{x\to\infty} f(x)
\]
\end{document}
```
[Open this example in Overleaf](https://www.overleaf.com/docs?engine=lualatex\&snip_name=Integrals%2C+Sums+and+Limits\&snip=%5Cdocumentclass%7Barticle%7D%0A%5Ctitle%7BIntegrals%2C+Sums+and+Limits%7D%0A%5Cauthor%7BOverleaf%7D%0A%5Cdate%7B%7D%0A%5Cusepackage%7Bamsmath%7D%0A%0A%5Cbegin%7Bdocument%7D%0A%0A%5Cmaketitle%0A%0A%5Csection%7BIntegrals%7D%0A%0AIntegral+%5C%28%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+dx%5C%29+inside+text.%0A%0A%5Cmedskip%0A%0AThe+same+integral+on+display%3A%0A%5C%5B%0A++++%5Cint_%7Ba%7D%5E%7Bb%7D+x%5E2+%5C%2Cdx%0A%5C%5D%0Aand+multiple+integrals%3A%0A%5Cbegin%7Bgather%2A%7D%0A++++%5Ciint_V+%5Cmu%28u%2Cv%29+%5C%2Cdu%5C%2Cdv%0A%5C%5C%0A++++%5Ciiint_V+%5Cmu%28u%2Cv%2Cw%29+%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Ciiiint_V+%5Cmu%28t%2Cu%2Cv%2Cw%29+%5C%2Cdt%5C%2Cdu%5C%2Cdv%5C%2Cdw%0A%5C%5C%0A++++%5Cidotsint_V+%5Cmu%28u_1%2C%5Cdots%2Cu_k%29+%5C%2Cdu_1+%5Cdots+du_k%0A%5C%5C%0A++++%5Coint_V+f%28s%29+%5C%2Cds%0A%5Cend%7Bgather%2A%7D%0A%0A%5Csection%7BSums+and+products%7D%0A%0ASum+%5C%28%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%5C%29+inside+text.%0A%0AThe+same+sum+on+display%3A%0A%5C%5B%0A++++%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+2%5E%7B-n%7D+%3D+1%0A%5C%5D%0A%0AProduct+%5C%28%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%5C%29+inside+text.%0A%0AThe+same+product+on+display%3A%0A%5C%5B%0A++++%5Cprod_%7Bi%3Da%7D%5E%7Bb%7D+f%28i%29%0A%5C%5D%0A%0A%5Csection%7BLimits%7D%0A%0ALimit+%5C%28%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%5C%29+inside+text.%0A%0AThe+same+limit+on+display%3A%0A%5C%5B%0A++++%5Clim_%7Bx%5Cto%5Cinfty%7D+f%28x%29%0A%5C%5D%0A%0A%5Cend%7Bdocument%7D)
## Further reading
For more information see
* [Mathematical expressions](/latex/mathematics/01-mathematical-expressions.md)
* [Subscripts and superscripts](/latex/mathematics/02-subscripts-and-superscripts.md)
* [Fractions and Binomials](/latex/mathematics/05-fractions-and-binomials.md)
* [Display style in math mode](/latex/mathematics/10-display-style-in-math-mode.md)
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