定积分的性质

(1)当$a=b$时，$\int_a^b {f(x){\text{d}}x} = 0$，这是因为此时无穷小量${\text{d}}x = 0$.

(2)当$a > b$时，$\int_a^b {f(x){\text{d}}x} = \int_b^a {f(x){\text{d}}x} $，这是因为此时定义中的小区间长度分别为

$$\eqalign{ & \Delta {x_1} = {x_1} - {x_0},\Delta {x_2} = {x_2} - {x_1}, \ldots , \cr & \Delta {x_n} = {x_n} - {x_{n - 1}} \cr & \Delta {x_1} = {x_0} - {x_1},\Delta {x_2} = {x_1} - {x_2}, \ldots , \cr & \Delta {x_n} = {x_{n - 1}} - {x_n} \cr} $$

除了特别说明，下列性质中的积分上、下限都不加以限制，不一定是$a < b$.

性质

$$\int_a^b {[f\left( x \right) \pm g\left( x \right)]{\text{d}}x} $$ $$= \int_a^b {f(x){\text{d}}x} \pm \int_a^b {g(x){\text{d}}x} $$

这是因为

$$\eqalign{ & \int_a^b {\left[ {f\left( x \right) \pm g\left( x \right)} \right]{\text{d}}x} \cr & = \mathop {\lim }\limits_{\lambda \to 0} \mathop \sum \limits_{i = 1}^n \left[ {f\left( {{\varrho _i}} \right) \pm g\left( {{\varrho _i}} \right)} \right]\Delta {x_i} \cr & = \mathop {\lim }\limits_{\lambda \to 0} \mathop \sum \limits_{i = 1}^n f\left( {{\varrho _i}} \right)\Delta {x_i} + \mathop {\lim }\limits_{\lambda \to 0} \mathop \sum \limits_{i = 1}^n g\left( {{\varrho _i}} \right)\Delta {x_i} \cr & = \int_a^b {f(x){\text{d}}x} \pm \int_a^b {g(x){\text{d}}x} \cr} $$

性质

$$\int_a^b {kf(x){\text{d}}x} = k\int_a^b {f(x){\text{d}}x},(k是常数) $$

性质

设$a < c < b$，则

$$\int_a^b {f(x){\text{d}}x} = \int_a^c {f(x){\text{d}}x} + \int_c^b {f(x){\text{d}}x{\text{}}} $$

性质

在区间$[a,b]$上，如果函数$f(x)≥0$，则

$$\int_a^b {f(x){\text{d}}x} \geqslant 0{\text{}}(a < b)$$

性质

在区间$[a,b]$上，如果函数$f(x)≤g(x)$，则

$$\int_a^b {f(x){\text{d}}x} \leqslant \int_a^b {g(x){\text{d}}x} {\text{}}(a < b)$$

性质

$$\left| {\int_a^b {f(x){\text{d}}x} } \right| \leqslant \int_a^b {|f(x)|{\text{d}}x} $$

性质

设$M,m$分别是函数$f(x)$在区间$[a,b]$上的最大值和最小值，则

$$m(b - a) \leqslant \int_a^b {f(x){\text{d}}x} \leqslant \int_a^b {M{\text{d}}x} $$

性质如果函数$f(x)$在积分区间$[a,b]$上连续，则在区间$[a,b]$上至少存在一个点$\varrho $，有

$$\int_a^b {f(x){\text{d}}x} = f(\varrho )(b - a){\text{}}(a \leqslant \varrho \leqslant b)$$