Calculus 2nd Edition Rogawski

SECTION 12.4 Area and Arc Length in Polar Coordinates 645
y
y
r = sin 2 θ
P
π
r = 4 sec( θ − 4
)
x
FIGURE 13
Q
x
FIGURE 17 Four-petaled rose r = sin 2θ.
10. Find the area enclosed by one loop of the lemniscate with equation
r 2 = cos 2θ (Figure 18). Choose your limits of integration carefully.
5. Find the area of the shaded region in Figure 14. Note that θ varies
from 0 to π 2 .
6. Which interval of θ-values corresponds to the the shaded region in
Figure 15? Find the area of the region.
y
−1 1
y
FIGURE 18 The lemniscate r 2 = cos 2θ.
11. Sketch the spiral r = θ for 0 ≤ θ ≤ 2π and find the area bounded
by the curve and the first quadrant.
x
8
r = θ
2 + 4θ
2
y
r = 3 − θ
12. Find the area of the intersection of the circles r = sin θ and
r = cos θ.
13. Find the area of region A in Figure 19.
y
r = 4 cos θ
1 2
FIGURE 14
x
FIGURE 15
3
x
r = 1
−1 1 2 4
A
x
7. Find the total area enclosed by the cardioid in Figure 16.
y
FIGURE 19
14. Find the area of the shaded region in Figure 20, enclosed by the
circle r = 1 2 and a petal of the curve r = cos 3θ. Hint: Compute the
area of both the petal and the region inside the petal and outside the
circle.
−2
−1
x
y
FIGURE 16 The cardioid r = 1 − cos θ.
r = cos 3θ
x
r = 1 2
8. Find the area of the shaded region in Figure 16.
9. Find the area of one leaf of the “four-petaled rose” r = sin 2θ (Figure
17). Then prove that the total area of the rose is equal to one-half
the area of the circumscribed circle.
FIGURE 20
15. Find the area of the inner loop of the limaçon with polar equation
r = 2 cos θ − 1 (Figure 21).