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Problems

Problems

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

Figure %: Polar Plot of 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:


(cos(2θ))2=  
 =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 sin2(θ) = (1 - cos(2θ))/2.

The cardioid looks like
Figure %: Polar Plot of r(θ) = sin(θ/2) for θ = 0 to 2Π
The area is equal to


sin2=  
 =θ - sin(θ))  
 =  

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