这一节，我们利用定积分的定义，得到定积分的一些性质。
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，\(M=2\)，\(m=1\)
\(\displaystyle\ \Longrightarrow\ 1\cdot (4-1)\leq \int_1^4 \sqrt xdx\leq 2\cdot (4-1)\)
\(\displaystyle\ \Longrightarrow\ 3\leq \int_1^4 \sqrt xdx\leq 6\)