Properties of the Definite Integral

Properties 6 and 7 relate the values of integrals of sums and differences of functions to the sums and differences of integrals of the individual functions. These two properties will be very useful when we need the integral of a function which is the sum or difference of several terms: we can integrate each term and then add or subtract the individual results to get the integral we want. Both of these new properties have natural interpretations as statements about areas of regions.

6. $\begin{aligned} &\int_{a}^{b} f(x)+g(x) \mathrm{dx}=\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x \end{aligned}$ (Fig. 3)

7. $\begin{aligned} &\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x)-\mathrm{g}(x) \mathrm{d} \mathrm{x}=\int_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}(x) \mathrm{d} \mathrm{x}-\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{g}(x) \mathrm{d} \mathrm{x} \\\end{aligned}$

Justification of Property 6: $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x)+\mathrm{g}(x) \mathrm{dx}$

$=\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n}\left(f\left(c_{k}\right)+g\left(c_{k}\right)\right) \Delta x_{k}\right)$

$=\lim _{\operatorname{mes} h \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}+\sum_{k=1}^{n} g\left(c_{k}\right) \Delta x_{k}\right)$

$=\lim _{\text {mesh } \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \cdot \Delta x_{k}\right)+\lim _{m e s h \rightarrow 0}\left(\sum_{k=1}^{n} g\left(c_{k}\right) \cdot \Delta x_{k}\right)=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} \mathrm{x}+\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{g}(x) \mathrm{dx}$.

Practice 1: $\int_{1}^{4} \mathrm{f}(x) \mathrm{dx}=7$, and $\int_{1}^{4} g(x) \mathrm{d} \mathrm{x}=3$. Evaluate $\int_{1}^{4} \mathrm{f}(x)-\mathrm{g}(x) \mathrm{d} \mathrm{x}$.

Property 8 says that if one function is larger than another function on an interval, then the definite integral of a larger function on that interval is bigger than the definite integral of the smaller function. This property then leads to Property 9 which provides a quick method for determining bounds on how large and small a particular integral can be.

8. If $\mathrm{f}(x) \leq \mathrm{g}(x)$ for all $x$ in $[a,b]$, then $$\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{dx} \leq \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{g}(x) \mathrm{dx}$. (Fig. 4)

9. $\text { (b-a) } \cdot(\min \text { of } f \text { on }[a, b]) \leq \int_{\mathrm{a}}^{\mathrm{b}} f(x) d x \leq(b-a) \cdot(\max \text { of } f \text { on }[a, b])$. (Fig. 5)

Justification of Property 8:

Fig. 4 illustrates that if $f$ and $g$ are both positive and $f(x) \leq \mathrm{g}(x)$ for all $x$ in $[a,b]$, then the area of region $F$ is smaller than the area of region $G$ and $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{d} \mathrm{x} \leq \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{g}(x) \mathrm{dx}$.

Similar sketches for the situations when $f$ or $g$ are sometimes or always negative illustrate that Property 9 is always true, but we can avoid all of the different cases by using Riemann sums.

Using Riemann Sums: If the same partition and sampling points $\mathrm{c}_{\mathrm{k}}$ are used to get Riemann sums for $f$ and $g$, then $\mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \leq \mathrm{g}\left(\mathrm{c}_{\mathrm{k}}\right)$ for each $k$ and $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}} \leq \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{g}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}} \text { so } \lim _{\text {mesh } \rightarrow 0}\left(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x_{k}\right) \leq \lim _{\operatorname{mesh} \rightarrow 0}\left(\sum_{k=1}^{n} g\left(c_{k}\right) \Delta x_{k}\right)$.

Justification of Property 9: Property 9 follows easily from Property 8.

Let $g(x)=M=(\max \text { of } f \text { on }[a, b])$. Then $f(x) \leq M=g(x)$ for all $x$ in $[a,b]$ so $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{dx} \leq \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{g}(\mathrm{x}) \mathrm{dx}=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{M} \mathrm{dx}=(\mathrm{b}-\mathrm{a}) \cdot \mathrm{M}$.

Example 1: Determine lower and upper bounds for the value of $\int_{1}^{5} f(x) \mathrm{d} \mathrm{x}$ in Fig. 6.

Solution: If $1 \leq x \leq 5$, then $2 \leq f(x) \leq 9$ so a lower bound is $(\mathrm{b}-\mathrm{a}) \cdot(\mathrm{min} \text { of } \mathrm{f} \text { on }[\mathrm{a}, \mathrm{b}])=(4)(2)=8$.

An upper bound is $(\mathrm{b}-\mathrm{a}) \cdot(\max \text { of } \mathrm{f} \text { on }[\mathrm{a}, \mathrm{b}])=(4)(9)=36: 8 \leq \int_{1}^{5} \mathrm{f}(x) \mathrm{d} \mathrm{x} \leq 36$. This range, from 8 to 36 is rather wide. Property 9 is not useful for finding the exact value of the integral, but it is very easy to use and it can help us avoid an unreasonable value for an integral.

Practice 2: Determine a lower bound and an upper bound for the value of $\int_{3}^{5} f(x) d x$ in Fig. 6.