定积分的几何意义

${\int }_a^{b}f(x)dx={\displaystyle \underset{d\to 0}{}\sum _{k=1}^{n}f(\xi k)\mathrm{\Delta }{x}_k}\backslash int\_\{a\}^\{b\}\; f(x)\; d\; x\; =\; \backslash displaystyle\backslash lim\_\{d\; \backslash to\; 0\}\; \backslash sum\_\{k\; =\; 1\}^\{n\}\; f(\backslash xi\; k)\; \backslash Delta\; x\_\{k\}$

提'可爱因子' $\frac{1}{n}\backslash frac\{1\}\{n\}$

不等式： $\text{ 若 }f(x)⩽g(x)\Rightarrow {\int }_a^{b}f(x)dx⩽{\int }_a^{b}g(x)dx(a\le b)\backslash text\; \{\; 若\; \}\; f(x)\; \backslash leqslant\; g(x)\; \backslash Rightarrow\; \backslash int\_\{a\}^\{b\}\; f(x)\; d\; x\; \backslash leqslant\; \backslash int\_\{a\}^\{b\}\; g(x)\; d\; x\; \backslash quad(a\; \backslash leq\; b)\backslash \backslash$

定积分的性质

变上限积分

定积分的计算

几何意义

反常积分 !!

无穷区间上的反常积分

定义1: ${\int }_a^{+\mathrm{\infty }}f(x)dx={\displaystyle \underset{t\to +\mathrm{\infty }}{}{\int }_a^{t}f(x)dx}\backslash int\_\{a\}^\{+\backslash infty\}\; f(x)\; d\; x\; =\; \backslash displaystyle\backslash lim\_\{t\; \backslash to\; +\; \backslash infty\}\; \backslash int\_\{a\}^\{t\}\; f(x)\; d\; x$

定义2: ${\int }_{-\mathrm{\infty }}^{b}f(x)dx={\displaystyle \underset{t\to -\mathrm{\infty }}{}{\int }_t^{b}f(x)dx}\backslash int\_\{-\backslash infty\}^\{b\}\; f(x)\; d\; x=\; \backslash displaystyle\backslash lim\; \_\{t\; \backslash rightarrow-\backslash infty\}\; \backslash int\_\{t\}^\{b\}\; f(x)\; d\; x$

定义3: ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(x)dx={\int }_{-\mathrm{\infty }}^{0}f(x)dx+{\int }_0^{+\mathrm{\infty }}f(x)dx\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; f(x)\; d\; x=\backslash int\_\{-\backslash infty\}^\{0\}\; f(x)\; d\; x+\backslash int\_\{0\}^\{+\backslash infty\}\; f(x)\; d\; x$

无界函数的反常积分

定义1: 设点a为函数f(x)的瑕点

${\int }_a^{b}f(x)dx={\displaystyle \underset{t\to a^{+}}{}{\int }_t^{b}f(x)dx}\backslash int\_\{a\}^\{b\}\; f(x)\; d\; x=\backslash displaystyle\backslash lim\; \_\{t\; \backslash rightarrow\; a^\{+\}\}\; \backslash int\_\{t\}^\{b\}\; f(x)\; d\; x$

定义2: 设点b为函数f(x)的瑕点

${\int }_a^{b}f(x)dx={\displaystyle \underset{t\to b^{-}}{}{\int }_a^{t}f(x)dx}\backslash int\_\{a\}^\{b\}\; f(x)\; d\; x=\backslash displaystyle\backslash lim\; \_\{t\; \backslash rightarrow\; b^\{-\}\}\; \backslash int\_\{a\}^\{t\}\; f(x)\; d\; x$

定义3: 设点c为函数f(x)的瑕点 (a < c < b)

${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx\backslash int\_\{a\}^\{b\}\; f(x)\; d\; x=\backslash int\_\{a\}^\{c\}\; f(x)\; d\; x+\backslash int\_\{c\}^\{b\}\; f(x)\; d\; x$