高数 05.01 定积分的概念与性质

$\color{blue}{第五章 定积分}$

$\color{blue}{第五章 第一节 定积分的概念与性质}$

$\color{blue}{一、曲边梯形面积}$

$\color{blue}{二、定积分定义}$

$设函数f(x)定义在[a, b]上，若对[a, b]的任一种分法 \\\\a = x_0 < x_1 < x_2 < \cdots < x_n = b,令\Delta{x_i} = x_i - x_{i - 1}, 任取 \\\\ \xi_{i} \in [x_{i - 1}, x_{i}],只要\lambda = max_{1 \leq i \leq n}{ \lbrace \Delta{x_i} \rbrace } \rightarrow 0时,\sum_{i = 1}^{n}{f(\xi_{i})\Delta{x_i}} \\\\ 总趋于确定的极限I,则称此极限I为函数f(x)在区间 \\\\ [a, b]上的定积分,记作\int_{a}^{b}{f(x) dx}\\\\ 即\ \int_{a}^{b}{f(x) dx} = \lim_{\lambda \rightarrow 0}{\sum_{i = 1}^{n}{f(\xi_{i})\Delta{x_i}}}$

$a:积分下限\\ b:积分上限\\ f(x):被积函数\\ x:积分变量\\ f(x)dx:被积表达式\\ \sum_{i = 1}^{n}{f(\xi_{i})\Delta{x_i}}\\ 定积分仅与被积函数及积分区间有关,而与积分变量用什么字母表示无关$

定积分的几何意义：
$f(x) > 0,\int_{a}^{b}{f(x)dx} = A 曲边梯形的面积$
$f(x) < 0,\int_{a}^{b}{f(x)dx} = -A曲边梯形面积的负值$
$\int_{a}^{b}{f(x)dx}就是f(x)曲线在区间[a,b]上面积的代数和$

可积的充分条件：
$定理1.函数f(x)在[a, b]上连续\Longrightarrow f(x)在[a, b]可积$
$定理2.函数f(x)在[a, b]上有界,且只有有限个间断点\Longrightarrow f(x)在[a, b]可积$

$\color{blue}{三、定积分的性质}$

$1.\int_{a}^{b}{f(x) dx} = -\int_{b}^{a}{f(x) dx} \Longrightarrow \int_{a}^{a}{f(x)dx} = 0$
$2.\int_{a}^{b}{dx} = b - a$
$3.\int_{a}^{b}{kf(x)dx} = k\int_{a}^{b}{f(x)dx} \quad (k 为常数)$
$4.\int_{a}^{b}{[f(x) \pm g(x)] dx} = \int_{a}^{b}{f(x)dx} \pm \int_{a}^{b}{g(x)dx}$
$证:左端=\lim_{\lambda \rightarrow 0}{\sum_{i = 1}^{n}{[f(\xi_i) \pm g(\xi_i)] \Delta{x_i} } } \\ =\lim_{\lambda \rightarrow 0}{\sum_{i = 1}^{n}{f(\xi_i) \Delta{x_i}}} \pm \lim_{\lambda \rightarrow 0}{\sum_{i = 1}^{n}{g(\xi_i)\Delta{x_i} } } \\ = \int_a^b {f(x)dx} \pm \int_a^b {g(x)dx}$
$5.\int_a^b{f(x)dx} = \int_a^c{f(x)dx} + \int_c^b{f(x)dx}$
$证：$
$当a < c < b时,$
$因f(x)在[a, b]上可积,$
$所以在分割区间时,可以永远取c为分点,$
$\sum_{[a, b]}{f(\xi_i) \Delta{x_i}} = \sum_{[a,c]}{f(\xi_i)\Delta(x_i)} + \sum_{[c, b]}{f(\xi_i)\Delta{x_i}}$
$\downarrow 令 \lambda \rightarrow 0$
$\int_a^b{f(x)dx} = \int_a^c{f(x)dx} + \int_c^b{f(x)dx}$
$当a, b, c的相对位置任意时,例如a < b < c,则有$
$\int_a^c{f(x)dx} = \int_a^b{f(x)dx} + \int_b^c{f(x)dx}$
$\int_a^b{f(x)dx} = \int_a^c{f(x)dx} - \int_b^c{f(x)dx}$
$\int_a^b{f(x)dx} = \int_a^c{f(x)dx} + \int_c^b{f(x)dx}$

$6.若在[a, b]上f(x) \geq 0,则\int_a^b{f(x)dx} \geq 0.$
$证：$
$\because \sum_{i = 1}^{n}{f(\xi_i)\Delta{x_i}} \geq 0$
$\therefore \int_a^b{f(x)dx} = \lim_{\lambda \rightarrow 0}{\sum_{i = 1}^{n}{f(\xi_i)\Delta{x_i}}} \geq 0$
$推论1.若在[a, b]上f(x) \leq g(x),则\int_a^b{f(x)dx} \leq \int_a^b{g(x)dx}$
$推论2.|\int_a^b{f(x)dx}| \leq \int_a^b{|f(x)|dx} \quad (a < b)$
$证：\because -|f(x)| \leq f(x) \leq |f(x)|$
$\therefore -\int_a^b{|f(x)|dx} \leq \int_a^b{f(x)dx} \leq \int_a^b{|f(x)| dx}$
$即\quad |\int_a^b{f(x)dx}| \leq \int_a^b{|f(x)|dx}$

$7.设M = max_{[a, b]}{f(x)}, m = min_{[a, b]}{f(x)}, 则$
$m(b - a) \leq \int_a^b{f(x) dx} \leq M(b - a) \quad (积分估值定理)$

$8.积分中值定理:$
$若f(x) \in C[a, b],则至少存在一点\xi \in [a, b],使\int_a^b{f(x)dx} = f(\xi)(b - a)$
$证：设f(x)在[a, b]上的最小值与最大值分别为m, M,则由性质7可得$
$m \leq \dfrac{1}{b - a}\int_a^b{f(x)dx} \leq M$
$根据闭区间上连续函数介值定理,在[a, b]上至少存在一点\xi \in [a, b],使$
$f(\xi) = \dfrac{1}{b - a}\int_a^b{f(x)dx}$
$说明：$
$积分中值定理对 a < b 或 a >b 都成立.$
$可把\dfrac{\int_a^b{f(x)dx}}{b - a} = f(\xi)理解为f(x)在[a, b]上的平均值.$

内容小结：
1.定积分的定义
2.定积分的性质
3.积分中值定理