   Chapter 8.2, Problem 35E

Chapter
Section
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=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=r2x2y=r2x2 (3)

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

y=r2x2 (4)

Substitute r2x2 for y in Equation (2)

y=rr2x2=rrr2x2=0 (5)

Rewrite the Equation (5) as follows:

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

x=rr2x2 (6)

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

y=r2x2 (7)

Substitute r2x2 for y in Equation (2):

y=rr2x2=rr+r2x2=0 (8)

Rewrite the Equation (8) as follows:

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

x=r+r2x2 (9)

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

S1=ab2πx1+(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(r2x2)=ddx[(r2x2)12]=12(r2x2)121(2x)=xr2x2

Substitute xr2x2 for dydx,rr2x2 for x, 0 for a, and 1 for b in Equation (10).

S1=rr2π(rr2x2)1+(xr2x2)2dx=20r2π(rr2x2)1+(x2r2x2)dx=4π0r(rr2x2)(r2x2+x2r2x2)dx=4π0r(rr2x2)r2r2x2dx

S1=4π0r(rr2x2)rr2x2dx=4π0r(r2r2x2r)dx (11)

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

S2=ab2πx1+(dydx)2dx (12)

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

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