Problem :
Plot the polar curve given by r(θ) = cos(2θ) for θ = 0 to 2Π.

Problem :
What is the area contained within the region bounded by r(θ) = cos(2θ) from
θ = 0 to 2Π? You may use that cos^{2}(θ) = (1 + cos(2θ))/2.

We compute the area as follows:

(cos(2θ))^{2}dθ

=

dθ

=

θ +

=

,

exactly half the area of the unit circle in which it is contained!

Problem :
Find the area bounded by the graph of the cardioid defined by
r(θ) = sin(θ/2) for θ = 0 to 2Π, using the
identity sin^{2}(θ) = (1 - cos(2θ))/2.

The cardioid looks like
The area is equal to

sin^{2}dθ

=

dθ

=

θ - sin(θ))

=

once again equal to half the area of the unit circle in which the region is contained!