Chapter 8.2, Problem 35E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y = r.

To determine

To find: The area of the surface obtained by rotating the circle about the line y=r.

Explanation

Given information:

The equation of the circle is x2+y2=r2 about the line y=r.

Therefore, the circle is bounded between y=âˆ’r and y=r.

Calculation:

Show the equations of the circle and line.

x2+y2=r2 (1)

y=r (2)

Rewrite the Equation (1) as follows

x2+y2=r2y2=r2âˆ’x2y=r2âˆ’x2 (3)

Show the Equation of curve for upper semicircle which is in positive x-axis:

y=r2âˆ’x2 (4)

Substitute r2âˆ’x2 for y in Equation (2)

y=rr2âˆ’x2=rrâˆ’r2âˆ’x2=0 (5)

Rewrite the Equation (5) as follows:

For the circle, y=r when x=0, therefore

x=râˆ’r2âˆ’x2 (6)

Show the Equation of curve for lower semicircle which is in negative x-axis:

y=âˆ’r2âˆ’x2 (7)

Substitute âˆ’r2âˆ’x2 for y in Equation (2):

y=râˆ’r2âˆ’x2=rr+r2âˆ’x2=0 (8)

Rewrite the Equation (8) as follows:

For the circle, y=r when x=0, therefore

x=r+r2âˆ’x2 (9)

Calculate the area of the surface obtained by rotating the upper semi circle about the line y=r. using the relation:

S1=âˆ«ab2Ï€x1+(dydx)2dx (10)

Here, S1 is the area of the surface obtained by rotating the upper semicircle about the line y=r.

Differentiate both sides of Equation (4) with respect to x.

dydx=ddx(r2âˆ’x2)=ddx[(r2âˆ’x2)12]=12(r2âˆ’x2)12âˆ’1(âˆ’2x)=âˆ’xr2âˆ’x2

Substitute âˆ’xr2âˆ’x2 for dydx,râˆ’r2âˆ’x2 for x, 0 for a, and 1 for b in Equation (10).

S1=âˆ«âˆ’rr2Ï€(râˆ’r2âˆ’x2)1+(âˆ’xr2âˆ’x2)2dx=2âˆ«0r2Ï€(râˆ’r2âˆ’x2)1+(x2r2âˆ’x2)dx=4Ï€âˆ«0r(râˆ’r2âˆ’x2)(r2âˆ’x2+x2r2âˆ’x2)dx=4Ï€âˆ«0r(râˆ’r2âˆ’x2)r2r2âˆ’x2dx

S1=4Ï€âˆ«0r(râˆ’r2âˆ’x2)rr2âˆ’x2dx=4Ï€âˆ«0r(r2r2âˆ’x2âˆ’r)dx (11)

Calculate the area of the surface obtained by rotating the lower semicircle about the line y=r using the relation:

S2=âˆ«ab2Ï€x1+(dydx)2dx (12)

Here, S2 is the area of the surface obtained by rotating the lower semicircle about the line y=r

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started