Chapter 8.2, Problem 27E

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8th Edition

James Stewart

ISBN: 9781305270336

Textbook Problem

If the region
*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:

The region is rotated about *x*-axis.

The curve is bounded between

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.

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

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

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

Substitute *y*, 1 for *a,* and *b* in Equation (2).

Consider the value of the function

Consider the value of the function

Within the limits

Show the definition of comparison theorem as shown below:

Suppose *f* and *g* are continuous function with

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