# 定积分的性质

• $$\displaystyle\int_a^af(x)dx=0$$，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;

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：

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\displaystyle\ \Longrightarrow\ 3\leq \int_1^4 \sqrt xdx\leq 6$$