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 + | ||
= | , |
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
sin^{2} dθ | = | dθ | |
= | θ - sin(θ)) | ||
= |