> 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/zh-tw/shu-xue/09-integrals-sums-and-limits.md).
# 積分、總和與上下限
## 積分
積分表達式可以使用以下方式加入 `\int_{lower}^{upper}` 指令。
請注意，積分表達式在行內與顯示數學模式下看起來可能會有些不同。
| LaTeX 程式碼 | 輸出 |
| ---------------------------------- | -------------------------------------------------------------------------------------- |
| `文字中的積分 \(\int_{a}^{b} x^2 \,dx\)` |  |
| `\[ \int_{a}^{b} x^2 \,dx \]` |  |
## 多重積分
若要取得二重／三重／多重積分以及循環積分，你必須使用 `amsmath` 和 `esint` （用於循環積分）的套件。
| LaTeX 程式碼 | 輸出 |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- |
| `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
\[
\oint_V f(s) \,ds
\]
` |  |
## 總和與乘積
與積分類似，總和表達式可以使用以下方式加入 `\sum_{lower}^{upper}` 指令。
| LaTeX 程式碼 | 輸出 |
| ----------------------------------------- | ----------------------------------------------------------------------------------------- |
| `文字中的總和 $\sum_{n=1}^{\infty} 2^{-n} = 1$` |  |
| `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` |  |
以類似方式，你可以使用以下方式得到由一系列因子組成的乘積表達式 `\prod_{lower}^{upper}` 指令。
| LaTeX 程式碼 | 輸出 |
| ------------------------------- | ------------------------------------------------------------------------------------------- |
| `文字中的乘積 $\prod_{i=a}^{b} f(i)$` |  |
| `\[ \prod_{i=a}^{b} f(i) \]` |  |
## 極限
極限表達式可以使用以下方式加入 `\lim_{lower}` 指令。
| LaTeX 程式碼 | 輸出 |
| --------------------------------- | --------------------------------------------------------------------------------- |
| `文字中的極限 $\lim_{x\to\infty} f(x)$` |  |
| `\[ \lim_{x\to\infty} f(x) \]` |  |
## 範例
以下是一個 LaTeX 範例文件：
```latex
\documentclass{article}
\title{積分、總和與極限}
\author{Overleaf}
\date{}
\usepackage{amsmath}
\begin{document}
\maketitle
\section{積分}
積分 \(\int_{a}^{b} x^2 dx\) 於文字中。
\medskip
同一個積分的顯示式：
\[
\int_{a}^{b} x^2 \,dx
\]
以及多重積分：
\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{總和與乘積}
文字中的總和 \(\sum_{n=1}^{\infty} 2^{-n} = 1\) 。
同一個總和的顯示式：
\[
\sum_{n=1}^{\infty} 2^{-n} = 1
\]
文字中的乘積 \(\prod_{i=a}^{b} f(i)\) 。
同一個乘積的顯示式：
\[
\prod_{i=a}^{b} f(i)
\]
\section{極限}
文字中的極限 \(\lim_{x\to\infty} f(x)\) 。
同一個極限的顯示式：
\[
\lim_{x\to\infty} f(x)
\]
\end{document}
```
[在 Overleaf 中開啟這個範例](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)
## 延伸閱讀
更多資訊請參見
* [數學表達式](/latex/zh-tw/shu-xue/01-mathematical-expressions.md)
* [下標與上標](/latex/zh-tw/shu-xue/02-subscripts-and-superscripts.md)
* [分數與二項式](/latex/zh-tw/shu-xue/05-fractions-and-binomials.md)
* [數學模式中的顯示樣式](/latex/zh-tw/shu-xue/10-display-style-in-math-mode.md)
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