Short and Sweet Calculus

## 6.2Properties of the definite integral

In this section, we discuss some key properties of definite integrals that will help us evaluate definite integrals in an easy way.

1. $${\displaystyle \int_{a}^{b}[f(x)+g(x)]dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx}.$$

2. $${\displaystyle \int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)}dx,$$ where $$c$$ is a constant.
In particular $${\displaystyle \int_{a}^{b}-f(x)dx=-\int_{a}^{b}f(x)dx}.$$

3. $${\displaystyle \int_{a}^{b}cdx=c(b-a)}$$, where $$c$$ is a constant.

4. $${\displaystyle \int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx=\int_{a}^{b}f(x)dx},$$ either $$c$$ is between $$a$$ and $$b$$ or outside.

The truth of Properties (1)–(3) immediately follows from the definition of the definite integral and the summation properties, namely $[f(x_{1}^{*})+g(x_{1}^{*})]\Delta x+\cdots+[f(x_{n}^{*})+g(x_{n}^{*})]\Delta x=\left[f(x_{1}^{*})\Delta x+\cdots+f(x_{n}^{*})\Delta x\right]+\left[g(x_{1}^{*})\Delta x+\cdots+g(x_{n}^{*})\Delta x\right]$ and $cf(x_{1}^{*})\Delta x+\cdots+cf(x_{n}^{*})\Delta x=c\left[f(x_{1}^{*})\Delta x+\cdots+f(x_{n}^{*})\Delta x\right].$ The truth of Property (3) can be shown graphically (see Figure 6.4).

Example 6.3. Find $$\int_{8}^{9}f(x)dx$$, if we know that $$\int_{-1}^{9}f(x)dx=-2$$ and $$\int_{-1}^{8}f(x)dx=4$$.

Solution

Using property 4, we have $\underbrace{\int_{-1}^{9}f(x)dx}_{-2}=\underbrace{\int_{-1}^{8}f(x)dx}_{4}+\int_{8}^{9}f(x)dx.$ Therefore, $\int_{8}^{9}f(x)dx=-6.$

1. If $$f(x)\geq0$$ for $$a<x<b$$ then $\int_{a}^{b}f(x)dx\geq0.$

2. If $$f(x)\geq g(x)$$ for $$a<x<b$$ then $\int_{a}^{b}f(x)dx\geq\int_{a}^{b}g(x)dx.$

Property (5) says if the graph of $$f$$ does not lie below the $$x$$-axis, the signed area of the region lying between the curve $$y=f(x)$$ and the $$x$$-axis is nonnegative. The truth of property (6), because $$f(x)-g(x)\geq0$$, follows from Properties (5), (1), and (2).