Mark M. answered 10/22/16
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Retired math prof. Calc 1, 2 and AP Calculus tutoring experience.
The cardioid is traced out as θ varies from 0 to 2π.
Area enclosed by the cardioid
= ½∫(from 0 to 2π) r2dθ
= ½∫(from 0 to 2π) (8+8sinθ)2dθ
= ½∫(from 0 to 2π) (64 + 32sinθ + 64sin2θ)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 m2
Yukari G.
04/20/17