Parametric and Polar Curves

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

Solution for Problem 1 >>

Figure %: Polar Plot of r(θ) = cos(2θ) for θ = 0 to 2Π

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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 .

Solution for Problem 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!

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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 .

Solution for Problem 3 >>

The cardioid looks like

Figure %: Polar Plot of r(θ) = sin(θ/2) for θ = 0 to 2Π

The area is equal to

sin2 dθ	=	dθ
	=	θ - sin(θ))
	=

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

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