Chapter 8.2, Problem 27E

### 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

# If the region R = { ( x , y ) | x ≥ 1 ,     0 ≤ y ≤ 1 / x } is rotated about the x-axis, the volume of the resulting solid is finite (see Exercise 7.8.63). Show that the surface area is infinite. (The surface is shown in the figure and is known as Gabriel’s horn.)

To determine

To show: The surface area obtained by rotating the region about x-axis is infinite.

Explanation

Given information:

The region is shown as follows:

â„œ={(x,y)â€‰|â€‰xâ‰¥1,0â‰¤yâ‰¤1x}

The region is rotated about x-axis.

The curve is bounded between x=1 and x=âˆž.

The surface obtained by rotating the region about x-axis is called Gabrielâ€™s Horn.

The volume of the resulting solid is finite.

Calculation:

Show the equation of the curve.

y=1x (1)

Calculate the area of the surface obtained by rotating the curve about x-axis using the relation:

S=âˆ«ab2Ï€y1+(dydx)2dx (2)

Here, S is the area of the surface obtained by rotating the curve about x-axis and

aâ‰¤xâ‰¤b.

Differentiate both sides of Equation (1) with respect to x.

dydx=ddx(1x)=ddx(xâˆ’1)=âˆ’1xâˆ’2=âˆ’1x2

Substitute âˆ’1x2 for dydx, 1x for y, 1 for a, and âˆž for b in Equation (2).

S=âˆ«1âˆž2Ï€1x1+(âˆ’1x2)2dx=2Ï€âˆ«1âˆž1x1+1x4â€‰dx=2Ï€âˆ«1âˆž1xx4+1x4â€‰dx=2Ï€âˆ«1âˆžx4+1x3â€‰dx (3)

Consider the value of the function f(x)=x4+1x3

Consider the value of the function g(x)=1x.

Within the limits [1,âˆž]

x4+1x3>1xf(x)>g(x)

Show the definition of comparison theorem as shown below:

Suppose f and g are continuous function with f(x)â‰¥g(x)â‰¥0 for xâ‰¥a

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