we see that the desired area is twice the sum of two parts: the area of the circle swept
out by θ as it varies from 0 to plus the area of the cardioid swept out by a radius vector as θ varies
from Consequently
FIGURE N7–7
See also Questions 46 and 47 in the Practice Exercises.
BC ONLYEXAMPLE 4
Find the area enclosed by the cardioid r = 2(1 + cos θ).
SOLUTION: We graphed the cardioid on our calculator, using polar mode, in the window [−2,5] ×
[−3,3] with θ in [0,2π].
FIGURE N7–8
Using the symmetry of the curve with respect to the polar axis we write