Chapter 8.3, Problem 48E

a.

To determine

To show : that $xf\left(x\right)=\sin x,0\le x\le \pi$ ..

Explanation of Solution

Given information :

Let $f\left(x\right)=\{\begin{array}{l}\raisebox{1ex}{$\left(\sin x\right)$}\!\left/ \!\raisebox{-1ex}{$x$}\right.,0\le x\le \pi \\ 1,x=0\end{array}$

The graph provided in the question shown below:

  

Calculation :

We know that for $0\le x\le \pi$

  $f\left(x\right)=\frac{\sin x}{x}$

Hence,

  $\begin{array}{l}x·f\left(x\right)=x·\frac{\sin x}{x}\\ \Rightarrow \sin x\end{array}$

b.

To determine

To find : the volume of the solid about the y-axis.

Answer to Problem 48E

The volume of the solid is $4\pi$ .

Explanation of Solution

Given information :

Let $f\left(x\right)=\{\begin{array}{l}\raisebox{1ex}{$\left(\sin x\right)$}\!\left/ \!\raisebox{-1ex}{$x$}\right.,0\le x\le \pi \\ 1,x=0\end{array}$

The graph provided in the question shown below:

  

Calculation :

Use the cylindrical shell method to make this problem simple

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(2\pi \right)\left(x\right)}\left(\frac{\sin x}{x}\right)dx\\ \Rightarrow 4\pi \end{array}$