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calc II question: polar curves and vertex focus and directrix line?

Find the area enclosed by the polar curve r= 2-cosTheta.

Set up but do not evaluate.

1.)the integral for the area inside r=4cos2theta and outside r=2

2.)and the integral for the area common to both.

Area
= 2 * [(1/2) ∫(0 to π) r^2 dθ
= ∫(0 to π) (2 - cosθ)^2 dθ
Hope you can do it from here. Answer is 9π/2.

1)
Area
= 4 [∫(0 to π/6) ((4cos2θ)^2 - (2)^2) dθ]
= 8π/3 + 4√3

2)
Area common to both curves
= Area of 4 loops of r = 4 cos 2θ - above area
= 4 * 2 * (1/2) [∫(0 to π/4) (4cos2θ)^2 dθ - above area
= (1/2) π (4)^2) - above area
= 8π - (8π/3 + 4√3)
= 16π/3 - 4√3
[Ans. 1) edited after Rozeta53 pointed out the error. Answer 2) also changed.]

[Please check the calculations.]

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