Chapter 6.3, Problem 3E

a

To determine

To find: The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(u\right)du}$

Answer to Problem 3E

  $5$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$

Formula used : Use property ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}={\displaystyle \underset{a}{\overset{b}{\int }}f\left(u\right)du}$

Calculation : Hence,

  ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(u\right)du}={\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}$ ......... (1)

Substitute ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$ in equation (1)

  ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(u\right)du}=5$

Hence, ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(u\right)du}=5$

b.

To determine

To find: The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}\sqrt{3}f\left(z\right)dz}$

Answer to Problem 3E

  $5\sqrt{3}$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$

Formula used : Use property ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}={\displaystyle \underset{a}{\overset{b}{\int }}f\left(u\right)du}$ and ${\displaystyle \underset{a}{\overset{b}{\int }}af\left(x\right)dx}=a{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$

Calculation : Hence,

  $\begin{array}{c}{\displaystyle \underset{1}{\overset{2}{\int }}\sqrt{3}f\left(z\right)dz}=\sqrt{3}{\displaystyle \underset{1}{\overset{2}{\int }}f\left(z\right)dz}\\ =\sqrt{3}{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}\end{array}$

Substitute ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$

  $\begin{array}{c}\sqrt{3}{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=\sqrt{3}\cdot 5\\ =5\sqrt{3}\end{array}$

Hence, ${\displaystyle \underset{1}{\overset{2}{\int }}\sqrt{3}f\left(z\right)dz}=5$

c.

To determine

To find: The value of the ${\displaystyle \underset{2}{\overset{1}{\int }}f\left(t\right)dt}$

Answer to Problem 3E

  $-5$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$

Formula used : Use property ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}={\displaystyle \underset{a}{\overset{b}{\int }}f\left(u\right)du}$ and ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}=-{\displaystyle \underset{b}{\overset{a}{\int }}f\left(x\right)dx}$

Calculation : Hence,

  $\begin{array}{c}{\displaystyle \underset{2}{\overset{1}{\int }}f\left(t\right)dt}=-{\displaystyle \underset{1}{\overset{2}{\int }}f\left(t\right)dt}\\ =-{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}\end{array}$

Substitute ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$ in equation (1)

  $-{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=-5$

Hence, ${\displaystyle \underset{2}{\overset{1}{\int }}f\left(t\right)dt}=-5$

d.

To determine

To find: The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}\left[-f\left(x\right)\right]dx}$

Answer to Problem 3E

  $-5$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$

Formula used: Use property ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}={\displaystyle \underset{a}{\overset{b}{\int }}f\left(u\right)du}$ and ${\displaystyle \underset{a}{\overset{b}{\int }}\left[-f\left(x\right)\right]dx}=-{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}$

Calculation: Hence,

  ${\displaystyle \underset{1}{\overset{2}{\int }}\left[-f\left(x\right)\right]dx}=-{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}$ ....... (1)

Substitute ${\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=5$ in equation (1)

  $-{\displaystyle \underset{1}{\overset{2}{\int }}f\left(x\right)dx}=-5$

Hence, ${\displaystyle \underset{1}{\overset{2}{\int }}\left[-f\left(x\right)\right]dx}=-5$