> 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/overleaf-learn-latex-pt/matematica/09-integrals-sums-and-limits.md). # Integrais, somas e limites ## Integrais Uma expressão integral pode ser adicionada usando o `\int_{lower}^{upper}` comando. Note que a expressão integral pode parecer um pouco diferente nos modos matemáticos inline e display. | Código LaTeX | Saída | | --------------------------------------------- | -------------------------------------------------------------------------------------- | | `Integral \(\int_{a}^{b} x^2 \,dx\) no texto` | ![Example of an integral inside text](/files/58ca2f319d866d9fde9ac983969f641fec4b90b8) | | `\[ \int_{a}^{b} x^2 \,dx \]` | ![Example of an integral](/files/c4b8d0f8da47bf4692518cb680f5105dd6d57bf0) | ## Integrais múltiplas Para obter integrais duplas/triplas/múltiplas e integrais cíclicas, você deve usar `amsmath` e `esint` (para integrais cíclicas) pacotes. | Código LaTeX | Saída | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | | `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/5e08c131177271f7725f5d94bf24055c8f9198a2) | | `latex
\[
\oint_V f(s) \,ds
\]
` | ![Example of integrals](/files/6da3a4d52cf55d7237938e890577167b7dd6c5a1) | ## Somas e produtos Tal como a integral, uma expressão de soma pode ser adicionada usando o `\sum_{lower}^{upper}` comando. | Código LaTeX | Saída | | ------------------------------------------------ | ----------------------------------------------------------------------------------------- | | `Soma $\sum_{n=1}^{\infty} 2^{-n} = 1$ no texto` | ![Example of sum expression inside text](/files/b8a7af6a8fa74a203bb6bd194eaef693a0ba518d) | | `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` | ![Example of sum expression](/files/a1ae55e8f349cce9718ba49f9131015771405989) | De forma semelhante, você pode obter uma expressão com o produto de uma sequência de fatores usando o `\prod_{lower}^{upper}` comando. | Código LaTeX | Saída | | ----------------------------------------- | ------------------------------------------------------------------------------------------- | | `Produto $\prod_{i=a}^{b} f(i)$ no texto` | ![Example of limit expression inside text](/files/5e6e94102068c35b61996769ebb232dc24ef7dac) | | `\[ \prod_{i=a}^{b} f(i) \]` | ![Example of limit expression](/files/19e128c173e41f5897933d9519dc3f0a85d6e7f3) | ## Limites Uma expressão de limite pode ser adicionada usando o `\lim_{lower}` comando. | Código LaTeX | Saída | | ------------------------------------------ | --------------------------------------------------------------------------------- | | `Limite $\lim_{x\to\infty} f(x)$ no texto` | ![Example of limits inside text](/files/774a69ed511b7243c8f4f5528c77a8d9bfbee0b6) | | `\[ \lim_{x\to\infty} f(x) \]` | ![Example of limits](/files/cbbdfe0388c1f22fafba134a83d0f6493a42815c) | ## Exemplos Aqui está um exemplo de documento LaTeX: ```latex \documentclass{article} \title{Integrais, Somas e Limites} \author{Overleaf} \date{} \usepackage{amsmath} \begin{document} \maketitle \section{Integrais} Integral \(\int_{a}^{b} x^2 dx\) no texto. \medskip A mesma integral em destaque: \[ \int_{a}^{b} x^2 \,dx \] e integrais múltiplas: \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{Somas e produtos} Soma \(\sum_{n=1}^{\infty} 2^{-n} = 1\) no texto. A mesma soma em destaque: \[ \sum_{n=1}^{\infty} 2^{-n} = 1 \] Produto \(\prod_{i=a}^{b} f(i)\) no texto. O mesmo produto em destaque: \[ \prod_{i=a}^{b} f(i) \] \section{Limites} Limite \(\lim_{x\to\infty} f(x)\) no texto. O mesmo limite em destaque: \[ \lim_{x\to\infty} f(x) \] \end{document} ``` [Abra este exemplo no 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) ## Leitura adicional Para mais informações, veja * [Expressões matemáticas](/latex/overleaf-learn-latex-pt/matematica/01-mathematical-expressions.md) * [Subscritos e sobrescritos](/latex/overleaf-learn-latex-pt/matematica/02-subscripts-and-superscripts.md) * [Frações e binómios](/latex/overleaf-learn-latex-pt/matematica/05-fractions-and-binomials.md) * [Estilo de destaque no modo matemático](/latex/overleaf-learn-latex-pt/matematica/10-display-style-in-math-mode.md) --- # Agent Instructions This documentation is published with GitBook. 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