Chapter 7.3, Problem 1E

a.

To determine

To find: The area of the cross-sections that are circular disks with diameters in the $xy$ -plane.

Answer to Problem 1E

  $\pi \left(1-x^2\right)$

Explanation of Solution

Given information:

The solid lies between the planes perpendicular to the x-axis at $x=-1$ and $x=1$ . The cross-sections perpendicular to x-axis between these planes run from semicircle $y=-\sqrt{1-x^2}$ to the semicircle $y=\sqrt{1-x^2}$ .

The cross-sections are circular disks with diameters in the $xy$ -plane.

Calculation:

Area of each circle is $\sqrt{1-x^2}$ .

So, the area of each cross-section is the area of the circle which is $\pi {\left(\sqrt{1-x^2}\right)}^2=\pi \left(1-x^2\right)$ .

Conclusion:

The area of the cross-section is $\pi \left(1-x^2\right)$ .

b.

To determine

To find: The area of the cross-sections that are squares with bases in the $xy$ -plane.

Answer to Problem 1E

  $4\left(1-x^2\right)$

Explanation of Solution

Given information:

The solid lies between the planes perpendicular to the x-axis at $x=-1$ and $x=1$ . The cross-sections perpendicular to x-axis between these planes run from semicircle $y=-\sqrt{1-x^2}$ to the semicircle $y=\sqrt{1-x^2}$ .

The cross-sections are squares with bases in the $xy$ -plane.

Calculation:

Base of each square is $2\sqrt{1-x^2}$ .

So, the area of each cross-section is the area of the square which is ${\left(2\sqrt{1-x^2}\right)}^2=4\left(1-x^2\right)$ .

Conclusion:

The area of the cross-section is $4\left(1-x^2\right)$ .

c.

To determine

To find: The area of the cross-sections that are squares with diagonals in the $xy$ -plane.

Answer to Problem 1E

  $2\left(1-x^2\right)$

Explanation of Solution

Given information:

The solid lies between the planes perpendicular to the x-axis at $x=-1$ and $x=1$ . The cross-sections perpendicular to x-axis between these planes run from semicircle $y=-\sqrt{1-x^2}$ to the semicircle $y=\sqrt{1-x^2}$ .

The cross-sections are squares with diagonals in the $xy$ -plane.

Calculation:

The diagonal of each square is $2\sqrt{1-x^2}$ . So, the base of each square is $\frac{2\sqrt{1-x^2}}{\sqrt{2}}=\sqrt{2}\sqrt{1-x^2}$ .

So, the area of each cross-section is the area of the square which is ${\left(\sqrt{2}\sqrt{1-x^2}\right)}^2=2\left(1-x^2\right)$ .

Conclusion:

The area of the cross-section is $2\left(1-x^2\right)$ .

d.

To determine

To find: The area of the cross-sections that are equilateral triangles with bases in the $xy$ -plane.

Answer to Problem 1E

  $\sqrt{3}\left(1-x^2\right)$

Explanation of Solution

Given information:

The solid lies between the planes perpendicular to the x-axis at $x=-1$ and $x=1$ . The cross-sections perpendicular to x-axis between these planes run from semicircle $y=-\sqrt{1-x^2}$ to the semicircle $y=\sqrt{1-x^2}$ .

The cross-sections are equilateral triangles with bases in the $xy$ -plane.

Calculation:

The base of each equilateral triangle is $2\sqrt{1-x^2}$ , and the height is $\sqrt{3}\sqrt{1-x^2}$ .

So, the area of each cross-section is the area of the equilateral triangle which is $\frac{1}{2}\times \sqrt{3}\sqrt{1-x^2}\times 2\sqrt{1-x^2}=\sqrt{3}\left(1-x^2\right)$ .

Conclusion:

The area of the cross-section is $\sqrt{3}\left(1-x^2\right)$ .