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# 将积分定义为黎曼和的极限

## "无限" 矩阵的黎曼和

${\int }_{2}^{6}\frac{1}{5}{x}^{2}\phantom{\rule{0.167em}{0ex}}dx$

$\underset{n\to \mathrm{\infty }}{lim}R\left(n\right)$

（这些事实的严格证明过程过于详细，无法在这里讲述。但没关系，我们只是通过直觉也可以看到黎曼和与定积分的关系。）

$\begin{array}{rl}\mathrm{\Delta }x& =\frac{6-2}{n}=\frac{4}{n}\\ \\ {x}_{i}& =2+\mathrm{\Delta }x\cdot i=2+\frac{4}{n}i\\ \\ f\left({x}_{i}\right)& =\frac{1}{5}\left({x}_{i}{\right)}^{2}=\frac{1}{5}{\left(2+\frac{4}{n}i\right)}^{2}\end{array}$

$R\left(n\right)=\sum _{i=1}^{n}{\left(2+\frac{4i}{n}\right)}^{2}\cdot \frac{4}{5n}$

$\begin{array}{rl}& \phantom{=}{\int }_{2}^{6}\frac{1}{5}{x}^{2}\phantom{\rule{0.167em}{0ex}}dx\\ \\ & =\underset{n\to \mathrm{\infty }}{lim}R\left(n\right)\\ \\ & =\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}{\left(2+\frac{4i}{n}\right)}^{2}\cdot \frac{4}{5n}\end{array}$

## 定积分是黎曼和的极限

${\int }_{a}^{b}f\left(x\right)dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}\mathrm{\Delta }x\cdot f\left({x}_{i}\right)$

## 给出定积分，如何写出黎曼和...

${\int }_{\pi }^{2\pi }\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$

$\begin{array}{rl}\mathrm{\Delta }x& =\frac{b-a}{n}\\ \\ & =\frac{2\pi -\pi }{n}\\ \\ & =\frac{\pi }{n}\end{array}$

$\begin{array}{rl}{x}_{i}& =a+\mathrm{\Delta }x\cdot i\\ \\ & =\pi +\frac{\pi }{n}\cdot i\\ \\ & =\pi +\frac{\pi i}{n}\end{array}$

${\int }_{\pi }^{2\pi }\mathrm{cos}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}\frac{\pi }{n}\cdot \mathrm{cos}\left(\pi +\frac{\pi i}{n}\right)$

### 练习根据定积分写出黎曼和

${\int }_{0}^{3}{e}^{x}\phantom{\rule{0.167em}{0ex}}dx=\phantom{\rule{0.167em}{0ex}}?$

${\int }_{1}^{e}\mathrm{ln}x\phantom{\rule{0.167em}{0ex}}dx=\phantom{\rule{0.167em}{0ex}}?$

## 根据黎曼和极限，写出定积分...

$\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}\mathrm{ln}\left(2+\frac{5i}{n}\right)\cdot \frac{5}{n}$

$\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}\mathrm{ln}\left(2+\frac{5i}{n}\right)\cdot \frac{5}{n}$

${\int }_{2}^{7}\mathrm{ln}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$

### 练习根据黎曼和，写定积分

$\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}{\left(3+\frac{4i}{n}\right)}^{2}\cdot \frac{4}{n}$

### 最后的常见难点： 不会分析表达式

$\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}\sqrt{4+\frac{5i}{n}}\cdot \frac{5}{n}=\phantom{\rule{0.167em}{0ex}}?$