Chapter 8.2, Problem 39E

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8th Edition

James Stewart

ISBN: 9781305270336

Textbook Problem

Formula 4 is valid only when *f*(*x*) ≥ 0. Show that when *f*(*x*) is not necessarily positive, the formula for surface area becomes

To determine

**To show:** The formula for surface area is

Explanation

**Given Information:**

The function

The area of the surface obtained by rotating the curve about *x*-axis is given by relation.

Here, the value of

**Calculation:**

Consider a frustum of cone with top radius
*l*.

Show the frustum of cone as shown in Figure1.

Refer to Figure 1.

Show the surface area of the frustum of cone (shaded region) using the relation.

Here, the average radius of the shaded region,

Consider the value of a curve

Here,

The curve
*x*-axis as shown in Figure 2:

Show the surface obtained by rotating the curve
*x*-axis as shown in Figure 3:

Refer to Figure 3.

The function *f* is a continuous function.

The point

The part of surface between
*x*-axis.

The resulting surface is a band (shaded area) as shown in Figure 3.

The radius of the band,

The slant height of the band,

Calculate the slant height of the band using the relation:

Here,

Consider the value of

The value of

The value of

Modify Equation (1).

Substitute
*r* and
*l* in Equation (1).

Modify Equation (2)

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