Chapter 6, Problem 9RE

(a)

To determine

To check statement

  ${\displaystyle {\int }_5^{2}f\left(x\right)}dx=-3$

is true or false.

Answer to Problem 9RE

Statement

  ${\displaystyle {\int }_5^{2}f\left(x\right)}dx=-3$

is a true statement.

Explanation of Solution

Given:

  $\begin{array}{l}{\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx=4\mathrm{......}\left(1\right)\\ {\displaystyle {\int }_2^{5}f\left(x\right)}dx=3\mathrm{.....}\left(2\right)\\ {\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx=2\mathrm{......}\left(3\right)\end{array}$

Calculation:

By property of definite integration

  $\begin{array}{l}{\displaystyle {\int }_5^{2}f\left(x\right)}dx=-{\displaystyle {\int }_2^{5}f\left(x\right)}dx\\ =-3\left(\text{from}\text{equation}\left(\text{2}\right)\right)\end{array}$

(b)

To determine

To check statement

  ${\displaystyle {\int }_{-2}^2\left[f\left(x\right)+g\left(x\right)\right]}dx=9$

is true or false.

Answer to Problem 9RE

Statement

  ${\displaystyle {\int }_{-2}^2\left[f\left(x\right)+g\left(x\right)\right]}dx=9$

is a true statement.

Explanation of Solution

Given:

  $\begin{array}{l}{\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx=4\mathrm{......}\left(1\right)\\ {\displaystyle {\int }_2^{5}f\left(x\right)}dx=3\mathrm{.....}\left(2\right)\\ {\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx=2\mathrm{......}\left(3\right)\end{array}$

Calculation:

Use property of definite integration

  $\begin{array}{l}{\displaystyle {\int }_a^b\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle {\int }_a^{b}f\left(x\right)}dx+{\displaystyle {\int }_a^{b}g\left(x\right)}dx\\ {\displaystyle {\int }_{-2}^5\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle {\int }_{-2}^{5}f\left(x\right)}dx+{\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx\mathrm{.......}\left(4\right)\end{array}$

Now use property of definite integration

  $\begin{array}{l}{\displaystyle {\int }_a^{b}f\left(x\right)}dx={\displaystyle {\int }_a^{c}f\left(x\right)}dx+{\displaystyle {\int }_c^{b}f\left(x\right)}dx\\ \therefore {\displaystyle {\int }_{-2}^{5}f\left(x\right)}dx={\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx+{\displaystyle {\int }_2^{5}f\left(x\right)}dx\mathrm{.....}\left(5\right)\end{array}$

From equation $\left(4\right)$ and $\left(5\right)$

  ${\displaystyle {\int }_{-2}^5\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx+{\displaystyle {\int }_2^{5}f\left(x\right)}dx+{\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx$

Now from equation $\left(1\right)$ , $\left(2\right)$ and $\left(3\right)$

  $\begin{array}{l}{\displaystyle {\int }_{-2}^5\left[f\left(x\right)+g\left(x\right)\right]}dx=4+3+2\\ =9\end{array}$

(c)

To determine

Answer to Problem 9RE

Statement $f\left(x\right)\le g\left(x\right)$ on interval $\left[-25\right]$ is false.

Explanation of Solution

Given:

  $\begin{array}{l}{\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx=4\mathrm{......}\left(1\right)\\ {\displaystyle {\int }_2^{5}f\left(x\right)}dx=3\mathrm{.....}\left(2\right)\\ {\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx=2\mathrm{......}\left(3\right)\end{array}$

Calculation:

By property of definite integration

If $f\left(x\right)\le g\left(x\right)$ on $\left[a,b\right]$ , then

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx\le {\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)}dx$

Now on interval $\left[-25\right]$

  $\begin{array}{l}{\displaystyle {\int }_{-2}^5\left[f\left(x\right)+g\left(x\right)\right]}dx={\displaystyle {\int }_{-2}^{2}f\left(x\right)}dx+{\displaystyle {\int }_2^{5}f\left(x\right)}dx\\ =4+3\\ =7\end{array}$

And

  $\begin{array}{l}{\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx=2\\ \therefore {\displaystyle {\int }_{-2}^{5}f\left(x\right)}dx\ge {\displaystyle {\int }_{-2}^{5}g\left(x\right)}dx\end{array}$

Thus

  $f\left(x\right)\ge g\left(x\right)$ on $\left[-2,5\right]$