Mark M. answered • 10/22/16

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The cardioid is traced out as θ varies from 0 to 2π.

Area enclosed by the cardioid

= ½∫

_{(from 0 to}_{2π)}r^{2}dθ = ½∫

_{(from 0 to 2π)}(8+8sinθ)^{2}dθ = ½∫

_{(from 0 to 2π)}(64 + 32sinθ + 64sin^{2}θ)dθ = 16∫

_{(from 0 to 2π) }(2+ sinθ + 2 [(1-cos(2θ))/2])dθ = 16∫

_{(from 0 to 2π)}(3 + sinθ - cos(2θ))dθ = 16[3θ - cosθ - ½(sin(2θ))]

_{(from 0 to 2π)} = 16[(6π-1-0) - (1)] = 16(6π-2) = 96π - 32 ≈ 269.6 m

^{2}
Yukari G.

^{2}^{2}θ^{2}+128sinθ + 6404/20/17