> 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/it/matematica/09-integrals-sums-and-limits.md). # Integrali, somme e limiti ## Integrali Un'espressione integrale può essere aggiunta usando il `\int_{lower}^{upper}` comando. Nota che un'espressione integrale può sembrare un po' diversa in modalità matematica inline e display. | Codice LaTeX | Risultato | | ----------------------------------------------------------- | -------------------------------------------------------------------------------------- | | `Integrale \(\int_{a}^{b} x^2 \,dx\) all'interno del testo` | ![Example of an integral inside text](/files/68ce9ef6d6169f392bb6bda37dc4b7f0011242e0) | | `\[ \int_{a}^{b} x^2 \,dx \]` | ![Example of an integral](/files/c3db1095753c80648c884cafb92c8c91aa9a8911) | ## Integrali multipli Per ottenere integrali doppi/tripli/multipli e integrali ciclici devi usare `amsmath` e `esint` (per gli integrali ciclici) pacchetti. | Codice LaTeX | Risultato | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | | `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/2062626d288a3d8cd4a2b441f2f0dea76079bb6a) | | `latex
\[
\oint_V f(s) \,ds
\]
` | ![Example of integrals](/files/3f013a546ac37c5833d244711e9983001d3d7633) | ## Somme e prodotti Come per gli integrali, un'espressione di somma può essere aggiunta usando il `\sum_{lower}^{upper}` comando. | Codice LaTeX | Risultato | | -------------------------------------------------------------- | ----------------------------------------------------------------------------------------- | | `Somma $\sum_{n=1}^{\infty} 2^{-n} = 1$ all'interno del testo` | ![Example of sum expression inside text](/files/34edfdbd530721aedcfa029567fc0d92903a0971) | | `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` | ![Example of sum expression](/files/780a0f495d64f66312e77b14b5bc0b84efff63ea) | In modo simile puoi ottenere un'espressione con il prodotto di una sequenza di fattori usando il `\prod_{lower}^{upper}` comando. | Codice LaTeX | Risultato | | ------------------------------------------------------- | ------------------------------------------------------------------------------------------- | | `Prodotto $\prod_{i=a}^{b} f(i)$ all'interno del testo` | ![Example of limit expression inside text](/files/a3ce9f927a747be60bb38244146ed8745919f55f) | | `\[ \prod_{i=a}^{b} f(i) \]` | ![Example of limit expression](/files/73b8002c018cf1785d59b6046e4ae62c543c7870) | ## Limiti Un'espressione di limite può essere aggiunta usando il `\lim_{lower}` comando. | Codice LaTeX | Risultato | | ------------------------------------------------------- | --------------------------------------------------------------------------------- | | `Limite $\lim_{x\to\infty} f(x)$ all'interno del testo` | ![Example of limits inside text](/files/3f6a781a29348da21b6a69ba1741e7b0a24eeff3) | | `\[ \lim_{x\to\infty} f(x) \]` | ![Example of limits](/files/9243a41990c4885e8c9c867d9c25ced14f03321d) | ## Esempi Ecco un documento LaTeX di esempio: ```latex \documentclass{article} \title{Integrals, Sums and Limits} \author{Overleaf} \date{} \usepackage{amsmath} \begin{document} \maketitle \section{Integrals} Integrale \(\int_{a}^{b} x^2 dx\) all'interno del testo. \medskip Lo stesso integrale in display: \[ \int_{a}^{b} x^2 \,dx \] e integrali multipli: \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} Somma \(\sum_{n=1}^{\infty} 2^{-n} = 1\) all'interno del testo. La stessa somma in display: \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \] Prodotto \(\prod_{i=a}^{b} f(i)\) all'interno del testo. Lo stesso prodotto in display: \[ \prod_{i=a}^{b} f(i) \] \section{Limits} Limite \(\lim_{x\to\infty} f(x)\) all'interno del testo. Lo stesso limite in display: \[ \lim_{x\to\infty} f(x) \] \end{document} ``` [Apri questo esempio 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) ## Approfondimenti Per maggiori informazioni vedi * [Espressioni matematiche](/latex/it/matematica/01-mathematical-expressions.md) * [Pedici e apici](/latex/it/matematica/02-subscripts-and-superscripts.md) * [Frazioni e binomi](/latex/it/matematica/05-fractions-and-binomials.md) * [Stile display in modalità matematica](/latex/it/matematica/10-display-style-in-math-mode.md) --- # Agent Instructions This documentation is published with GitBook. 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