Chapter 8.3, Problem 29E

a.

To determine

To find : the volume of the solid.

Answer to Problem 29E

The volume of the solid is $8\pi$ .

Explanation of Solution

Given information : The region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about the x-axis.

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{4}{\int }}\pi \left(4-x\right)dx}\\ \Rightarrow 8\pi \end{array}$

b.

To determine

To find : the volume of the solid.

Answer to Problem 29E

The volume of the solid is $\frac{32\pi }{5}$ .

Explanation of Solution

Given information : The region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about the y-axis.

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{2}{\int }}\pi y^{4}dy}\\ \Rightarrow \frac{32\pi }{5}\end{array}$

c.

To determine

To find : the volume of the solid.

Answer to Problem 29E

The volume of the solid is $\frac{8\pi }{3}$ .

Explanation of Solution

Given information : The region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about the line $y=2$ .

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{4}{\int }}\pi {\left(2-\sqrt{x}\right)}^{2}dx}\\ \Rightarrow \frac{8\pi }{3}\end{array}$

d.

To determine

To find : the volume of the solid.

Answer to Problem 29E

The volume of the solid is $\frac{224\pi }{15}$ .

Explanation of Solution

Given information : The region bounded by $y=\sqrt{x}$ and the lines $y=2$ and $x=0$ about the line $x=4$ .

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{2}{\int }}16\pi -\pi {\left(4-y^2\right)}^{2}dy}\\ \Rightarrow \frac{224\pi }{15}\end{array}$