> 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/es/matematicas/09-integrals-sums-and-limits.md). # Integrales, sumas y límites ## Integrales La expresión integral se puede agregar usando `\int_{lower}^{upper}` comando. Nota que la expresión integral puede parecer un poco diferente en el modo matemático en línea y en el modo de visualización. | Código LaTeX | Salida | | ----------------------------------------------------- | -------------------------------------------------------------------------------------- | | `Integral \(\int_{a}^{b} x^2 \,dx\) dentro del texto` | ![Example of an integral inside text](/files/4771c137a682505d502a00e3365b3ef50086d5f0) | | `\[ \int_{a}^{b} x^2 \,dx \]` | ![Example of an integral](/files/8c30a302a5a934d5646b643c725227aadfbf08c3) | ## Integrales múltiples Para obtener integrales dobles/triples/múltiples e integrales cíclicas debes usar `amsmath` y `esint` (para integrales cíclicas) paquetes. | Código LaTeX | Salida | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | | `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*}
` | ![Example of multiple integrals](/files/1dc4a719ce1858b745ec26d39348eba8cb8dc508) | | `latex
\[
\oint_V f(s) \,ds
\]
` | ![Example of integrals](/files/cb1e7aa8bc05cac30a44c28fd779bb3fc15a92f0) | ## Sumas y productos Al igual que la integral, la expresión de suma se puede agregar usando `\sum_{lower}^{upper}` comando. | Código LaTeX | Salida | | -------------------------------------------------------- | ----------------------------------------------------------------------------------------- | | `Suma $\sum_{n=1}^{\infty} 2^{-n} = 1$ dentro del texto` | ![Example of sum expression inside text](/files/856d29ddd472e1b0ecea22f31aa76b2a602f9dc0) | | `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` | ![Example of sum expression](/files/aa507d13082e3639ec069655272811154fed2612) | De manera similar, puedes obtener una expresión con el producto de una secuencia de factores usando `\prod_{lower}^{upper}` comando. | Código LaTeX | Salida | | -------------------------------------------------- | ------------------------------------------------------------------------------------------- | | `Producto $\prod_{i=a}^{b} f(i)$ dentro del texto` | ![Example of limit expression inside text](/files/5397930f93b705d54117c668e690ee0ce7cdc03e) | | `\[ \prod_{i=a}^{b} f(i) \]` | ![Example of limit expression](/files/72038082005252f57a5ce089cbe2ff3f92d3c06b) | ## Límites La expresión de límite se puede agregar usando `\lim_{lower}` comando. | Código LaTeX | Salida | | -------------------------------------------------- | --------------------------------------------------------------------------------- | | `Límite $\lim_{x\to\infty} f(x)$ dentro del texto` | ![Example of limits inside text](/files/cd3b99d06ca3ce3a03978a4c91a1f0382377da91) | | `\[ \lim_{x\to\infty} f(x) \]` | ![Example of limits](/files/1be5a723d8b76d9c96d2415b7b2d6e00ade76c9f) | ## Ejemplos Aquí hay un documento de ejemplo en LaTeX: ```latex \documentclass{article} \title{Integrales, sumas y límites} \author{Overleaf} \date{} \usepackage{amsmath} \begin{document} \maketitle \section{Integrales} Integral \(\int_{a}^{b} x^2 dx\) dentro del texto. \medskip La misma integral en formato de visualización: \[ \int_{a}^{b} x^2 \,dx \] y múltiples integrales: \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{Sumas y productos} Suma \(\sum_{n=1}^{\infty} 2^{-n} = 1\) dentro del texto. La misma suma en formato de visualización: \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \] Producto \(\prod_{i=a}^{b} f(i)\) dentro del texto. El mismo producto en formato de visualización: \[ \prod_{i=a}^{b} f(i) \] \section{Límites} Límite \(\lim_{x\to\infty} f(x)\) dentro del texto. El mismo límite en formato de visualización: \[ \lim_{x\to\infty} f(x) \] \end{document} ``` [Abre este ejemplo en 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) ## Lectura adicional Para más información, consulta * [Expresiones matemáticas](/latex/es/matematicas/01-mathematical-expressions.md) * [Subíndices y superíndices](/latex/es/matematicas/02-subscripts-and-superscripts.md) * [Fracciones y binomios](/latex/es/matematicas/05-fractions-and-binomials.md) * [Estilo de visualización en modo matemático](/latex/es/matematicas/10-display-style-in-math-mode.md) --- # Agent Instructions This documentation is published with GitBook. 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