   Chapter 8.2, Problem 27E

Chapter
Section
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:

={(x,y)|x1,0y1x}

The region is rotated about x-axis.

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

The surface obtained by rotating the region about x-axis is called Gabriel’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=ab2πy1+(dydx)2dx (2)

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

axb.

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

dydx=ddx(1x)=ddx(x1)=1x2=1x2

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

S=12π1x1+(1x2)2dx=2π11x1+1x4dx=2π11xx4+1x4dx=2π1x4+1x3dx (3)

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

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

Within the limits [1,]

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)g(x)0 for xa

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