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

Section 8.2 Area of a Surface of Revolution 553
With the Leibniz notation for derivatives, this formula becomes
5
S − y b
a
2yÎ 1 1 S dy
dxD2
dx
If the curve is described as x − tsyd, c < y < d, then the formula for surface area
becomes
6
S − y d
c
2yÎ 1 1 S dx
dyD2
dy
and both Formulas 5 and 6 can be summarized symbolically, using the notation for arc
length given in Section 8.1, as
7
S − y 2y ds
For rotation about the y-axis, the surface area formula becomes
8
S − y 2x ds
where, as before, we can use either
ds −Î1 1S
dxD
dy
2
dx or ds −Î1 1S
dyD
dx 2
dy
These formulas can be remembered by thinking of 2y or 2x as the circumference of a
circle traced out by the point sx, yd on the curve as it is rotated about the x-axis or y-axis,
respectively (see Figure 5).
y
y
(x, y)
0
y
x
x
(x, y)
FIGURE 5
circumference=2πy
(a) Rotation about x-axis: S=j 2πy ds
circumference=2πx
0
x
(b) Rotation about y-axis: S=j 2πx ds
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.