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Parametric and Polar Curves

Problems

The Area Below a Polar Curve

How to Cite This SparkNote

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!

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