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

654 Chapter 10 Parametric Equations and Polar Coordinates
we have
L − y
2
0
ÎS
dD
dx
2
1S dy
dD2
d
− y 2
sr 2 s1 2 cos d 2 1 r 2 sin 2 d
0
The result of Example 5 says that the
length of one arch of a cycloid is eight
times the radius of the gener ating circle
(see Figure 5). This was first proved
in 1658 by Sir Christopher Wren, who
later became the architect of St. Paul’s
Cathedral in London.
y
0
r
L=8r
2πr
x
− y 2
sr 2 s1 2 2 cos 1 cos 2 1 sin 2 d d
0
− r y 2
s2s1 2 cos d d
0
To evaluate this integral we use the identity sin 2 x − 1 2 s1 2 cos 2xd with − 2x, which
gives 1 2 cos − 2 sin 2 sy2d. Since 0 < < 2, we have 0 < y2 < and so
sinsy2d > 0. Therefore
and so
s2s1 2 cos d − s4 sin 2 sy2d − 2 | sinsy2d | − 2 sinsy2d
L − 2r y 2
sinsy2d d − 2rf22 cossy2dg 2
0
0
FIGURE 5
− 2rf2 1 2g − 8r
n
Surface Area
In the same way as for arc length, we can adapt Formula 8.2.5 to obtain a formula for surface
area. Suppose the curve c given by the parametric equations x − f std, y − tstd, < t < ,
where f 9, t9 are continuous, tstd > 0, is rotated about the x-axis. If C is traversed exactly
once as t increases from to , then the area of the resulting surface is given by
6
S − y
2yÎS
dtD
dx
2
1S
dtD
dy
2
dt
The general symbolic formulas S − y 2y ds and S − y 2x ds (Formulas 8.2.7 and
8.2.8) are still valid, but for parametric curves we use
ds −ÎS
dtD
dx
2
1S
dtD
dy
2
dt
Example 6 Show that the surface area of a sphere of radius r is 4r 2 .
SOLUTION The sphere is obtained by rotating the semicircle
x − r cos t y − r sin t 0 < t <
about the x-axis. Therefore, from Formula 6, we get
S − y
2r sin t ss2r sin 0 td2 1 sr cos td 2 dt
− 2 y
r sin t sr 2 ssin 2 t 1 cos 2 td dt − 2 y
r sin t ? r dt
0 0
− 2r 2 y
0 sin t dt − 2r 2 s2cos tdg 0
− 4r
2
n
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.