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

672 Chapter 10 Parametric Equations and Polar Coordinates
so, using cos 2 1 sin 2 − 1, we have
S
dD
dx 2
1S
dD
dy 2
−S
dD
dr 2
cos 2 2 2r dr
d cos sin 1 r 2 sin 2
1 S
dD
dr 2
sin 2 1 2r dr
d sin cos 1 r 2 cos 2
−S
dD
dr 2
1 r 2
Assuming that f 9 is continuous, we can use Theorem 10.2.5 to write the arc length as
b
L − y ÎS dx 2
a dD 1S
dD
dy
2
d
Therefore the length of a curve with polar equation r − f sd, a < < b, is
5
L − y
b
a
Îr 2 1S
dD
dr
2
d
O
Example 4 Find the length of the cardioid r − 1 1 sin .
SOLUTION The cardioid is shown in Figure 8. (We sketched it in Example 10.3.7.) Its
full length is given by the parameter interval 0 < < 2, so Formula 5 gives
L − y
2
0
Îr 2 1S
dD
dr
2
d − y 2
ss1 1 sin d 2 1 cos 2 d − y 2
s2 1 2 sin d
0
0
FIGURE 8
r − 1 1 sin
We could evaluate this integral by multiplying and dividing the integrand by
s2 2 2 sin , or we could use a computer algebra system. In any event, we find that
the length of the cardioid is L − 8.
n
1–4 Find the area of the region that is bounded by the given
curve and lies in the specified sector.
1. r − e 2y4 , y2 < <
5–8 Find the area of the shaded region.
5.
6.
2. r − cos , 0 < < y6
3. r − sin 1 cos , 0 < <
4. r − 1y, y2 < < 2
r@= sin 2¨
r=2+cos ¨
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.