Chapter 7.3, Problem 5E

To determine

To find: The volume of the solid.

Answer to Problem 5E

The answer: $\frac{16}{3}$

Explanation of Solution

Given information:

The $x$ -axis at:

  $\begin{array}{c}x=-1\\ x=1\end{array}$

The cross section perpendicular to the $x$ -axis between these planes are square whose bases run from the semi circle $y=-\sqrt{1-x^2}$ to the semi circle

  $y=\sqrt{1-x^2}$ .

Calculation:

Utilise the area function created for Exercise $1(b)$ since this problem is a continuation of that exercise, obtained an area function of $A(x)=4\left(1-x^2\right)$ .

The bounds of the region are $x=-1$ and

  $x=1$ , so integrate the area function from $-1$ to $1$ to find the volume.

  

  $\begin{array}{c}={\left.4\left(x-\frac{1}{3}{x}^3\right)\right|}_{-1}^1\\ =4\left(1-\frac{1}{3}{(1)}^3\right)-4\left(-1-\frac{1}{3}{(-1)}^3\right)\\ =4\left(1-\frac{1}{3}\right)-4\left(-1+\frac{1}{3}\right)\\ =4\left(\frac{2}{3}\right)-4\left(-\frac{2}{3}\right)\\ =\frac{8}{3}+\frac{8}{3}\\ =\frac{16}{3}\end{array}$

Therefore, the required volume of the solid is $\frac{16}{3}$ .