> 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/cs/matematika/09-integrals-sums-and-limits.md). # Integrály, součty a limity ## Integrály Integrální výraz lze přidat pomocí `\int_{lower}^{upper}` příkaz. Všimněte si, že integrální výraz může v řádkovém a zobrazeném matematickém režimu vypadat trochu jinak. | Kód LaTeX | Výstup | | -------------------------------------------- | -------------------------------------------------------------------------------------- | | `Integrál \(\int_{a}^{b} x^2 \,dx\) v textu` | ![Example of an integral inside text](/files/61aee6d19613d2395382912996f7f106a250c839) | | `\[ \int_{a}^{b} x^2 \,dx \]` | ![Example of an integral](/files/abac644dd9aa6b239fcbdb54dc695c8ed29032b3) | ## Vícenásobné integrály Pro získání dvojitých/trojitých/vícenásobných integrálů a cyklických integrálů musíte použít `amsmath` a `esint` (pro cyklické integrály) balíčky. | Kód LaTeX | Výstup | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | | `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/50becc73d25e639b7ca5eb961f3da7a427ebf13f) | | `latex
\[
\oint_V f(s) \,ds
\]
` | ![Example of integrals](/files/461c9359dc40a62a52306ee80b975b00f8a34db8) | ## Součty a součiny Podobně jako integrál lze výraz pro součet přidat pomocí `\sum_{lower}^{upper}` příkaz. | Kód LaTeX | Výstup | | ------------------------------------------------- | ----------------------------------------------------------------------------------------- | | `Součet $\sum_{n=1}^{\infty} 2^{-n} = 1$ v textu` | ![Example of sum expression inside text](/files/67efe2aa3aed58e338f392b87d358d4428feab81) | | `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` | ![Example of sum expression](/files/c8dd05d124d6061e31a9b2ae0b5fb76e4a2b8b43) | Podobným způsobem můžete získat výraz s produktem posloupnosti faktorů pomocí `\prod_{lower}^{upper}` příkaz. | Kód LaTeX | Výstup | | --------------------------------------- | ------------------------------------------------------------------------------------------- | | `Součin $\prod_{i=a}^{b} f(i)$ v textu` | ![Example of limit expression inside text](/files/d5609381cb8621f863aadbea28babbc8277b3c43) | | `\[ \prod_{i=a}^{b} f(i) \]` | ![Example of limit expression](/files/4977225c0155ab6bd1fd4d2bb9c7e1f223bb8399) | ## Limity Výraz pro limitu lze přidat pomocí `\lim_{lower}` příkaz. | Kód LaTeX | Výstup | | ----------------------------------------- | --------------------------------------------------------------------------------- | | `Limita $\lim_{x\to\infty} f(x)$ v textu` | ![Example of limits inside text](/files/6ea3a6841262198b4a92fd6852d6eb2a279ba4ba) | | `\[ \lim_{x\to\infty} f(x) \]` | ![Example of limits](/files/3d769e2d305dcd6e78028b356dd0b8015507024b) | ## Příklady Zde je ukázkový dokument LaTeX: ```latex \documentclass{article} \title{Integrály, součty a limity} \author{Overleaf} \date{} \usepackage{amsmath} \begin{document} \maketitle \section{Integrály} Integrál \(\int_{a}^{b} x^2 dx\) v textu. \medskip Tentýž integrál v zobrazení: \[ \int_{a}^{b} x^2 \,dx \] a vícenásobné integrály: \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{Součty a součiny} Součet \(\sum_{n=1}^{\infty} 2^{-n} = 1\) v textu. Tentýž součet v zobrazení: \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \] Součin \(\prod_{i=a}^{b} f(i)\) v textu. Tentýž součin v zobrazení: \[ \prod_{i=a}^{b} f(i) \] \section{Limity} Limita \(\lim_{x\to\infty} f(x)\) v textu. Tentýž limit v zobrazení: \[ \lim_{x\to\infty} f(x) \] \end{document} ``` [Otevřete tento příklad v Overleafu](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) ## Další čtení Další informace najdete v * [Matematické výrazy](/latex/cs/matematika/01-mathematical-expressions.md) * [Dolní a horní indexy](/latex/cs/matematika/02-subscripts-and-superscripts.md) * [Zlomky a binomy](/latex/cs/matematika/05-fractions-and-binomials.md) * [Styl zobrazení v matematickém režimu](/latex/cs/matematika/10-display-style-in-math-mode.md) --- # Agent Instructions This documentation is published with GitBook. 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