Chapter 6, Problem 5RCC

To determine

To Represent: The integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$.

To Find: the unit of the integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$.

Answer to Problem 5RCC

The integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$ represents as the amount of work done.

The unit of the integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$ is $\underset{\_}{\text{N}\cdot \text{m}}$ or J.

Explanation of Solution

The amount of work done or energy is equals to the product of force and distance.

$\begin{array}{c}\text{Work}\text{done}=\text{Force}\times \text{distance}\\ =f\left(x\right)\times x\end{array}$

Here, $f\left(x\right)$ is force and x is distance.

Given information:

A book is pushed from 0 to 6 m with the force $f\left(x\right)$.

Calculation:

Consider the A book is pushed from 0 to 6 m with the force $f\left(x\right)$. Hence, the upper limit as 6 and lower limit as 0.

Express the integral function for work done to push the book from 0 to 6 m as follows:

$\text{Work}\text{done}={\displaystyle \underset{0}{\overset{6}{\int }}f\left(x\right)}dx$

Therefore, the integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$ represents as the amount of work done.

Find the unit of the integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$ as follows:

$\begin{array}{c}\text{Work}\text{done}=\text{Force}\times \text{distance}\\ =\text{N}\times \text{m}\\ =\text{N}\cdot \text{m}\end{array}$

Convert the unit from $\text{N}\cdot \text{m}$ to J.

$1\text{N}\cdot \text{m}=1\text{J}$

Therefore, the unit of the integral function ${\displaystyle {\int }_0^{6}f\left(x\right)dx}$ is $\underset{\_}{\text{N}\cdot \text{m}}$ or J.