> 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/ja/shu-xue/09-integrals-sums-and-limits.md).
# 積分、総和、極限
## 積分
積分式は次を使って追加できます `\int_{lower}^{upper}` コマンド。
なお、積分式はインライン数式モードと表示数式モードで少し異なって見える場合があります。
| LaTeXコード | 出力 |
| ---------------------------------- | -------------------------------------------------------------------------------------- |
| `本文中の積分 \(\int_{a}^{b} x^2 \,dx\)` |  |
| `\[ \int_{a}^{b} x^2 \,dx \]` |  |
## 重積分
二重積分／三重積分／多重積分および閉曲線積分を得るには、次を使用する必要があります `amsmath` や `esint` （閉曲線積分用）パッケージ。
| LaTeXコード | 出力 |
| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- |
| `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
\[
\oint_V f(s) \,ds
\]
` |  |
## 和と積
積分と同様に、和の式は次を使って追加できます `\sum_{lower}^{upper}` コマンド。
| LaTeXコード | 出力 |
| ---------------------------------------- | ----------------------------------------------------------------------------------------- |
| `本文中の和 $\sum_{n=1}^{\infty} 2^{-n} = 1$` |  |
| `\[ \sum_{n=1}^{\infty} 2^{-n} = 1 \]` |  |
同様に、次を使って一連の因子の積を含む式を得ることができます `\prod_{lower}^{upper}` コマンド。
| LaTeXコード | 出力 |
| ------------------------------ | ------------------------------------------------------------------------------------------- |
| `本文中の積 $\prod_{i=a}^{b} f(i)$` |  |
| `\[ \prod_{i=a}^{b} f(i) \]` |  |
## 極限
極限式は次を使って追加できます `\lim_{lower}` コマンド。
| LaTeXコード | 出力 |
| --------------------------------- | --------------------------------------------------------------------------------- |
| `本文中の極限 $\lim_{x\to\infty} f(x)$` |  |
| `\[ \lim_{x\to\infty} f(x) \]` |  |
## 例
以下は LaTeX 文書の例です:
```latex
\documentclass{article}
\title{積分、和、極限}
\author{Overleaf}
\date{}
\usepackage{amsmath}
\begin{document}
\maketitle
\section{積分}
本文中の積分 \(\int_{a}^{b} x^2 dx\)。
\medskip
同じ積分を表示すると:
\[
\int_{a}^{b} x^2 \,dx
\]
および重積分:
\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{和と積}
本文中の和 \(\sum_{n=1}^{\infty} 2^{-n} = 1\)。
同じ和を表示すると:
\[
\sum_{n=1}^{\infty} 2^{-n} = 1
\]
本文中の積 \(\prod_{i=a}^{b} f(i)\)。
同じ積を表示すると:
\[
\prod_{i=a}^{b} f(i)
\]
\section{極限}
本文中の極限 \(\lim_{x\to\infty} f(x)\)。
同じ極限を表示すると:
\[
\lim_{x\to\infty} f(x)
\]
\end{document}
```
[Overleaf でこの例を開く](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)
## さらに読む
詳細については
* [数式](/latex/ja/shu-xue/01-mathematical-expressions.md)
* [下付き文字と上付き文字](/latex/ja/shu-xue/02-subscripts-and-superscripts.md)
* [分数と二項係数](/latex/ja/shu-xue/05-fractions-and-binomials.md)
* [数式モードでの表示スタイル](/latex/ja/shu-xue/10-display-style-in-math-mode.md)
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