Chapter 8.3, Problem 31E

a.

To determine

To find : the volume of the solid.

Answer to Problem 31E

The volume of the solid is $\frac{16\pi }{15}$ .

Explanation of Solution

Given information : The region bounded by parabola $y=x^2$ and the line $y=1$ about the line $y=1$ .

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{-1}{\overset{1}{\int }}\pi {\left(1-x^2\right)}^{2}dx}\\ \Rightarrow \frac{16\pi }{15}\end{array}$

b.

To determine

To find : the volume of the solid.

Answer to Problem 31E

The volume of the solid is $\frac{56\pi }{15}$ .

Explanation of Solution

Given information : The region bounded by parabola $y=x^2$ and the line $y=1$ about the line $y=2$ .

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{-1}{\overset{1}{\int }}\pi {\left(2-x^2\right)}^2-\pi dx}\\ \Rightarrow \frac{56\pi }{15}\end{array}$

c.

To determine

To find : the volume of the solid.

Answer to Problem 31E

The volume of the solid is $\frac{64\pi }{15}$ .

Explanation of Solution

Given information : The region bounded by parabola $y=x^2$ and the line $y=1$ about the line $y=-1$ .

Calculation :

The volume of the solid:

  $\begin{array}{l}{\displaystyle \underset{-1}{\overset{1}{\int }}4\pi -\pi {\left(x^2+1\right)}^{2}dx}\\ \Rightarrow \frac{64\pi }{15}\end{array}$