Chapter 7.3, Problem 76E

To determine

To Find: The volume of a solid generated when R revolved about the x-axis.

Answer to Problem 76E

The volume of a solid is $128\prod$ which is option (B).

Explanation of Solution

Given information: The region R is enclosed by $y=e^{-x}$ , $y=e^x$ and $x=1$ .

Calculation:

The region R is revolved around x- axis. Now, find the intersection point first:

  $\begin{array}{l}y=e^{-x}=e^x\\ e^x-e^{-x}=0\\ e^x-\frac{1}{e^x}=0\\ e^{2x}-1=0\\ e^{2x}=1\\ 2x\ln e=\ln 1\\ 2x=0\\ x=0\end{array}$

The graph of $y=e^{-x}$ is exponential decay and $y=e^x$ is exponential growth. Hence, the area can be find out by subtracting the area of $y=e^x$ in the interval $0\le x\le 1$ from area of $y=e^{-x}$ in the interval $0\le x\le 1$

Hence, area can be find out as:

  $A={\displaystyle \underset{0}{\overset{1}{\int }}(e^x-e^{-x})dx}$

Now, volume of a solid is;

  $\begin{array}{l}V={\displaystyle \underset{a}{\overset{b}{\int }}A(x)dx}\\ V={\displaystyle \underset{0}{\overset{1}{\int }}(e^x-e^{-x})dx}\end{array}$

Therefore, option (a) ${\displaystyle \underset{0}{\overset{1}{\int }}(e^x-e^{-x})dx}$ is the correct option.