The two curves intersect where 1/2=cosθ. So θ=−π/3 or π/3.
The area we want is then
21∫−π/3π/3((cosθ)2−(1/2)2)dθ
=81∫−π/3π/3(2(1+cos(2θ))−1)dθ
=81∫−π/3π/3(1+2cos(2θ))dθ
=81[θ+sin(2θ)]π/3−π/3
=81(3π+sin(32π)−(−3π)−sin(3−2π))
=(12π+83)(units2) The area outside circle r=1/2 and inside circle r=cosθ is (12π+83) square units.
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