Chapter 8.2, Problem 39E

### 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

# Formula 4 is valid only when f(x) ≥ 0. Show that when f(x) is not necessarily positive, the formula for surface area becomes S = ∫ a b 2 π | f ( x ) | 1 + [ f ′ ( x ) ] 2   d x

To determine

To show: The formula for surface area is S=ab2π|f(x)|1+[f(x)]2dx , when f(x) is not necessarily positive.

Explanation

Given Information:

The function f(x) is not necessarily positive.

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

S=âˆ«ab2Ï€f(x)1+[fâ€²(x)]2dx

Here, the value of f(x)â‰¥0 .

Calculation:

Consider a frustum of cone with top radius r1 and bottom radius r2 and slanting height 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.

S=2Ï€rl (1)

Consider the value of a curve y=f(x) and it is a continuous function.

Here, aâ‰¤xâ‰¤b

The curve y=f(x) is rotated about x-axis as shown in Figure 2:

Show the surface obtained by rotating the curve y=f(x) about x-axis as shown in Figure 3:

Refer to Figure 3.

The function f is a continuous function.

The point Pi(xi,yi) lies on the curve.

The part of surface between xi and xiâˆ’1 is approximated by rotating the line segment PiPiâˆ’1 about x-axis.

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

The radius of the band, r=12(yiâˆ’1+yi) .

The slant height of the band, l=|Piâˆ’1Pi| .

Calculate the slant height of the band using the relation:

l=|Piâˆ’1Pi|=1+[fâ€²(xi*)]2Î”x

Here, xi* lies in the interval [xiâˆ’1,xi] .

Consider the value of Î”x to be very small and the function f(x) is not necessarily positive.

The value of yi=|f(xi)|â‰ˆ|f(xi*)| .

The value of yiâˆ’1=|f(xiâˆ’1)|â‰ˆ|f(xi*)| .

Modify Equation (1).

Substitute 12(yiâˆ’1+yi) for r and |Piâˆ’1Pi| for l in Equation (1).

S=2Ï€rl=2Ï€12(yiâˆ’1+yi)|Piâˆ’1Pi| (2)

Modify Equation (2)

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