Chapter 9, Problem 54RE

To determine

To find : The volume of the solid.

Answer to Problem 54RE

The volume of the solid is $2\pi$ .

Explanation of Solution

Given information :

The curve $y=-\ln x$ in the first quadrant is revolved about the x-axis to generate a solid.

Calculation :

Since, the curve is in first quadrant and revolved around the x-axis.

Therefore, volume of the solid so obtained is given by:

  $V={\displaystyle {\int }_0^{\infty }yxdy}$ .................... (1)

Hence,

  $\begin{array}{l}y=-\ln x\\ -y=\ln x\\ e^{-y}=x\\ x=e^{-y}\end{array}$

Substitute $x=e^{-y}$ in equation (1) and solve:

  $\begin{array}{l}V=2\pi {\displaystyle {\int }_0^{\infty }y{e}^{-y}dy}\\ V=2\pi \underset{a\to \infty }{\lim }{\displaystyle {\int }_0^{a}y{e}^{-y}dy}\\ V=2\pi \underset{a\to \infty }{\lim }{\left[-y{e}^{-y}\right]}_0^a+\underset{a\to \infty }{\lim }{\displaystyle {\int }_0^a{e}^{-y}dy}\\ V=2\pi \underset{a\to \infty }{\lim }{\left[-y{e}^{-y}-e^{-y}\right]}_0^a\\ V=2\pi \underset{a\to \infty }{\lim }\left[\left(-a{e}^{-a}-e^{-a}\right)-\left(-0e^{-0}-e^{-0}\right)\right]\\ V=2\pi \left[\left(0-0\right)-\left(0-1\right)\right]\\ V=2\pi \end{array}$

Hence,

The volume of the solid is $2\pi$ .