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 cos2(θ) = (1 + cos(2θ))/2.We compute the area as follows:
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 sin2(θ) = (1 - cos(2θ))/2.The cardioid looks like The area is equal to
|=||θ - sin(θ))|
once again equal to half the area of the unit circle in which the region is contained!