Properties of the Definite Integral

As you read each statement about definite integrals, examine the associated Figure and interpret the property as a statement about areas.

1. $\int_{\mathrm{a}}^{\mathrm{a}} \mathrm{f}(x) \mathrm{d} \mathrm{x}=0$                                    (a definition)

2. $\int_{b}^{a} f(x) d x=-\int_{a}^{b} f(x) d x$               (a definition)

3. $\int_{a}^{b} k d x=k \cdot(b-a)$                          (Fig. 1)

4. $\int_{a}^{b} k \cdot f(x) d x=k \cdot \int_{a}^{b} f(x) d x$

5. $\int_{a}^{b} f(x) d x+\int_{b}^{c} f(x) \mathrm{dx}=\int_{b}^{c} \mathrm{f}(x) \mathrm{d} x$              (Fig. 2)

Justification of Property 3: Using area: If $k>0$ then $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{k} \mathrm{dx}$ represents the area of the rectangle with $base = b–a$ and $height = k$, so $\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{k} \mathrm{d} \mathrm{x}=(\text { height }) \cdot(\text { base })=\mathrm{k}\cdot(\mathrm{~b}-\mathrm{a})$

Using Riemann Sums: For every partition $\mathrm{P}=\left\{x_{0}=\mathrm{a}, x_{1}, x_{2}, x_{3}, \ldots, x_{\mathrm{n}-1}, x_{\mathrm{n}}=\mathrm{b}\right\}$ of the interval [a, b], and every choice of $\mathrm{c}_{\mathrm{k}}$, the Riemann sum is

$\begin{aligned} \sum_{k=1}^{n} f\left(c_{k}\right) \cdot \Delta x_{k} &=\sum_{k=1}^{n} \mathrm{k} \cdot \Delta x_{k}=k \sum_{k=1}^{n} \Delta x_{k} \\ &=k \sum_{k=1}^{n}\left(x_{k}-x_{k-1}\right)=k\left\{\left(x_{1}-x_{0}\right)+\left(x_{2}-x_{1}\right)+\left(x_{3}-x_{2}\right)+\ldots+\left(x_{n-1}-x_{n}\right)\right\} \\ &=k \cdot\left(x_{n}-x_{0}\right)=k \cdot(b-a) \end{aligned}$

so the limit of the Riemann sums, as the mesh approaches zero, is $\mathrm{k} \cdot(\mathrm{b}-\mathrm{a}) . \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{k} \mathrm{d} x=\mathrm{k} \cdot(\mathrm{b}-\mathrm{a})$.

Justification of Property 4:

$\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{k} \cdot \mathrm{f}(x) \mathrm{dx}=\lim _{\operatorname{mesh} \varnothing 0}\left(\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{k} \cdot \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}}\right)=\mathrm{k} \cdot \lim _{\operatorname{mesh} \varnothing 0}\left(\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}\left(\mathrm{c}_{\mathrm{k}}\right) \Delta \mathrm{x}_{\mathrm{k}}\right)=\mathrm{k} \cdot \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(x) \mathrm{dx}$.

Property 5 can be justified using Riemann sums, but Fig. 2 graphically illustrates why it is true.