Chapter 8.3, Problem 69E

a.

To determine

To find : the value of $c$ and also find the minimum volume.

Answer to Problem 69E

The value of $c=\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.$ and the minimum volume is $\frac{{\pi }^2-8}{2}$ .

Explanation of Solution

Given information : The arch $y=\sin x,0\le x\le \pi$ , is revolved about the line $y=c,0\le c\le 1$, to generate the solid in figure:

  

Calculation :

To find the value of $c$ that minimizes the integral.

  ${\displaystyle \underset{0}{\overset{\pi }{\int }}\pi {\left(\sin \left(x\right)-c\right)}^{2}dx}$

Solving and taking the derivate, we get the minimizing value of $c$ as $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.$

This gives a volume of $\frac{{\pi }^2-8}{2}$

b.

To determine

To find : what value of $c$ in $\left[0,1\right]$

Answer to Problem 69E

The value of $c=1$

Explanation of Solution

Given information : The arch $y=\sin x,0\le x\le \pi$ , is revolved about the line $y=c,0\le c\le 1$ , to generate the solid in figure

  

Calculation :

The value of $c$ which maximizes the integral in (a) is $c=1$

c.

To determine

To graph : the solid’s volume as a function of $c$ first for $0\le c\le 1$

Explanation of Solution

Given information : The arch $y=\sin x,0\le x\le \pi$ , is revolved about the line $y=c,0\le c\le 1$ , to generate the solid in figure

  

Calculation :

The volume of the solid is given by the equation

  $V=\frac{\pi \left(2c^2\pi -8c+\pi \right)}{2}$

As the value of $c$ moves away from the interval $\left[0,1\right]$ , the volume goes to infinity