Chapter 8.3, Problem 47E

a.

To determine

To find : the volume of the solid by using the washer method.

Answer to Problem 47E

The volume of the solid is $\frac{11\pi }{48}$ .

Explanation of Solution

Given information : The region in the first quadrant that is bounded above by the curve $y=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{x}$}\right.$ , on the left by the line $x=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ , and below by the line $y=1$ is revolved about the y -axis to generate a solid.

Calculation :

By using the washer method.

  $\begin{array}{l}V=\pi {\displaystyle \underset{1}{\overset{2}{\int }}\left({\left(\frac{1}{y^2}\right)}^2-{\left(\frac{1}{4}\right)}^2\right)}dy\\ V=\pi {\displaystyle \underset{1}{\overset{2}{\int }}\left(\frac{1}{y^4}-\frac{1}{16}\right)}dy\\ V=\pi {\left(-\frac{1}{3y^3}-\frac{1}{16}y\right)}_1^2\\ V=\pi \left(-\frac{1}{24}-\frac{1}{8}-\left(-\frac{1}{3}-\frac{1}{16}\right)\right)\\ V=\frac{11\pi }{48}\end{array}$

b.

To determine

To find : the volume of the solid by using the cylindrical shell method.

Answer to Problem 47E

The volume of the solid is $\frac{11\pi }{48}$ .

Explanation of Solution

Given information : The region in the first quadrant that is bounded above by the curve $y=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{x}$}\right.$ , on the left by the line $x=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ , and below by the line $y=1$ is revolved about the y -axis to generate a solid.

Calculation :

By using the cylindrical shell method.

  $\begin{array}{l}V=2\pi {\displaystyle \underset{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\overset{1}{\int }}\left(x\left(\frac{1}{\sqrt{x}}-x\right)\right)}dx\\ V=2\pi {\displaystyle \underset{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\overset{1}{\int }}\left(\sqrt{x}-x\right)}dy\\ V=2\pi {\left(\frac{2}{3}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-\frac{1}{2}{x}^2\right)}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^1\\ V=2\pi \left(\frac{2}{3}-\frac{1}{2}-\left(\frac{1}{12}+\frac{1}{32}\right)\right)\\ V=\frac{11\pi }{48}\end{array}$