Chapter 6.3, Problem 5E

a.

To determine

To calculate : Find the value of the ${\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}$

Answer to Problem 5E

  $4$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{0}{\overset{3}{\int }}f\left(z\right)dz}=3$ and ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(z\right)dz}=7$

Formula used : Use property ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx}={\displaystyle \underset{a}{\overset{c}{\int }}f\left(x\right)dx}+{\displaystyle \underset{c}{\overset{b}{\int }}f\left(x\right)dx}$ , where $a\le c\le b$

Calculation : Hence, the integral ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(z\right)dz}$ e can be written as

  ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(z\right)dz}={\displaystyle \underset{0}{\overset{3}{\int }}f\left(z\right)dz}+{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}$   ...... (1)

Substitute ${\displaystyle \underset{0}{\overset{3}{\int }}f\left(z\right)dz}=3$ and ${\displaystyle \underset{0}{\overset{4}{\int }}f\left(z\right)dz}=7$ in equation (1)

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{4}{\int }}f\left(z\right)dz}={\displaystyle \underset{0}{\overset{3}{\int }}f\left(z\right)dz}+{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}\\ 7=3+{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}\end{array}$

Subtract $3$ from both sides and further simplify it

  $\begin{array}{l}7-3=3+{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}-3\\ 4={\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}\end{array}$

Hence, ${\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}=4$

b.

To determine

To calculate : Find the value of the ${\displaystyle \underset{4}{\overset{3}{\int }}f\left(t\right)dt}$

Answer to Problem 5E

  $-4$

Explanation of Solution

Given information : The value of the ${\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}=4$ from part(a)

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

Calculation : Hence,

  $\begin{array}{c}{\displaystyle \underset{4}{\overset{3}{\int }}f\left(t\right)dt}=-{\displaystyle \underset{3}{\overset{4}{\int }}f\left(t\right)dt}\\ =-{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}\end{array}$

Substitute ${\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}=4$ in equation (1)

  $-{\displaystyle \underset{3}{\overset{4}{\int }}f\left(z\right)dz}=-4$

Hence, ${\displaystyle \underset{4}{\overset{3}{\int }}f\left(t\right)dt}=-4$