定积分的性质

这一节,我们利用定积分的定义,得到定积分的一些性质。

笔记下载:定积分的性质

首先,由定积分的几何意义(曲边梯形的面积),我们规定两点:

  • \(\displaystyle\int_a^af(x)dx=0\),这是因为直线的面积为 \(0\);
  • \(\displaystyle\int_b^af(x)dx=-\int_a^bf(x)dx\),这是因为所有的 \(\Delta x_i\) 都取负号;

由定积分的定义(和式的极限),我们有下列的性质:

1,\(\displaystyle\ \int_a^b Cdx=C(b-a)\),这是因为长方形的面积等于底乘以高。

2,\(\displaystyle\ \int_a^b [pf(x)\pm qg(x)]dx=p\int_a^b f(x)dx\pm q\int_a^b g(x)dx\)

3,区域可加性:

\(\displaystyle c\in(a,b)\quad\Rightarrow\quad\int _a^b f(x)dx=\int_a^c f(x)dx+\int _c^b f(x)dx\)

例1,求\(\displaystyle\ \int_0^1 (4+3x^2)dx\)

解答:因\(\displaystyle\ \int_0^1 x^2dx=\frac{1}{3}\)

\(\displaystyle\ \int_0^1 (4+3x^2)dx=\int_0^1 4dx+3\int_0^1 x^2dx\)

\(\displaystyle\qquad\qquad\quad\quad\ \ \ \ =4\cdot 1+3\cdot \frac{1}{3}=5\)

例2,已知\(\displaystyle\ \int_0^{10} f(x)dx=17\),\(\displaystyle\ \int_0^8 f(x)dx=12\),求\(\displaystyle\ \int_8^{10} f(x)dx\)

解答:\(\displaystyle\ \int_0^{10} f(x)dx=\int_0^8 f(x)dx+\int_8^{10} f(x)dx\)

\(\displaystyle\Longrightarrow 17=12+\int_8^{10} f(x)dx\quad\Longrightarrow\quad \int_8^{10}f(x)dx=5\)

4,若\(\displaystyle\ f(x)\geq 0\),\(\displaystyle\forall x\in[a,b]\quad\Rightarrow\quad\int _a^b f(x)dx\geq 0\)

5,若\(\displaystyle\ f(x)\leq g(x)\),\(\displaystyle\forall x\in[a,b]\quad\Rightarrow\quad\int _a^b f(x)dx \leq \int_a^b g(x)dx\)

证明:\(\displaystyle \int_a^b g(x)dx-\int _a^b f(x)dx=\int_a^b[g(x)-f(x)]dx\geq 0\quad\Rightarrow\quad\int_a^b g(x)dx\geq \int _a^b f(x)dx\)

6,估值定理:若\(m\leq f(x)\leq M\),\(\displaystyle m(b-a)\leq\int_a^b f(x)dx\leq M(b-a)\)

证明:\(\displaystyle \int _a^b mdx\leq\int_a^b f(x)dx\leq\int_a^b Mdx\)

\(\displaystyle\Longrightarrow\ m(b-a)\leq \int_a^b f(x)dx\leq M(b-a)\)

7,积分中值定理:\(\displaystyle\ f(x)\)在\(\displaystyle\ [a,b]\ \)上连\(\displaystyle\ \Longrightarrow\ \)存在\(\displaystyle\ \xi \in (a,b)\),使得

\(\displaystyle\qquad\int _a^b f(x)dx=f(\xi)(b-1)\)

证明:\(\displaystyle\Longrightarrow\ m(b-a)\leq \int_a^b f(x)dx\leq M(b-a)\)

\(\displaystyle\qquad\ \ \Longrightarrow\ m\leq \frac{1}{b-a}\int _a^b f(x)dx\leq M\)

\(\displaystyle\qquad\ \Longrightarrow\ \)设\(\displaystyle\ C=\frac{1}{b-a}\int _a^b f(x)dx\),则\(\displaystyle C\in(m,M)\)

由介值定理,存在\(\displaystyle\ \xi \in (a,b)\),使得\(\displaystyle\ f(\xi)=C\)

\(\displaystyle\qquad\ \ \Longrightarrow\ f(\xi)=\frac{1}{b-a}\int _a^b f(x)dx\ \ \Longrightarrow\ \int_a^b f(x)dx=f(\xi)(b-a)\)

例3,估计\(\displaystyle\ \int_1^4 \sqrt xdx\ \) 的值

解答:在\(\ [1,4]\ \)区间上,\(M=2\),\(m=1\)

\(\displaystyle\ \Longrightarrow\ 1\cdot (4-1)\leq \int_1^4 \sqrt xdx\leq 2\cdot (4-1)\)

\(\displaystyle\ \Longrightarrow\ 3\leq \int_1^4 \sqrt xdx\leq 6\)