James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

552 Chapter 8 Further Applications of Integration
Putting this in Equation 1, we get
A − sr 1 l 1 r 2 ld
or
2
A − 2rl
y y=ƒ
0 a b
(a) Surface of revolution
x
where r − 1 2 sr 1 1 r 2 d is the average radius of the band.
Now we apply this formula to our strategy. Consider the surface shown in Figure 4,
which is obtained by rotating the curve y − f sxd, a < x < b, about the x-axis, where f
is positive and has a continuous derivative. In order to define its surface area, we divide
the interval fa, bg into n subintervals with endpoints x 0 , x 1 , . . . , x n and equal width Dx,
as we did in determining arc length. If y i − f sx i d, then the point P i sx i , y i d lies on the
curve. The part of the surface between x i21 and x i is approximated by taking the line
segment P i21 P i and rotating it about the x-axis. The result is a band with slant height
l − | P i21 P i | and average radius r − 1 2 sy i21 1 y i d so, by Formula 2, its surface area is
y
P¸
y i
P i-1
P i
P n
2 y i21 1 y i
2
| P i21 P i |
0 a
b x
(b) Approximating band
FIGURE 4
As in the proof of Theorem 8.1.2, we have
| P i21 P i | − s1 1 f f 9sx i*dg 2 Dx
where x i * is some number in fx i21 , x i g. When Dx is small, we have y i − f sx i d < f sx i *d and
also y i21 − f sx i21 d < f sx i *d, since f is continuous. Therefore
2 y i21 1 y i
2
| P i21 P i | < 2 f sx i*d s1 1 f f 9sx i *dg 2 Dx
and so an approximation to what we think of as the area of the complete surface of revolution
is
3
o n
2 f sx i *d s1 1 f f 9sx i *dg 2 Dx
i−1
This approximation appears to become better as n l ` and, recognizing (3) as a Riemann
sum for the function tsxd − 2 f sxd s1 1 f f 9sxdg 2 , we have
lim
n l ` on 2 f sx i *d s1 1 f f 9sx i *dg 2 Dx − y b
2 f sxd s1 1 f f 9sxdg 2 dx
i−1
a
Therefore, in the case where f is positive and has a continuous derivative, we define the
surface area of the surface obtained by rotating the curve y − f sxd, a < x < b, about
the x-axis as
4
S − y b
2 f sxd s1 1 f f 9sxdg 2 dx
a
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