Polar Coordinates DIFFICULT question! PLEASE HELP!!!

Question

When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front on the stage (as in the figure) and the boundary of the optimal pickup region is given by the given cardioid r = 8 + 8 sin θ, where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question. (Round your answer to two decimal places.)

Answer 1

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