Difficult cardioid/ Polar Graph Equation!!!

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 2 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, 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. Use CAS to answer their question. (Give your answer correct to two decimal places.)

r = 4 + 4sin(θ)

Answer 1

This is a simple problem. r dr dθ = dxdy. Thus, using I[a, b] for the integral from a to b and E[a,b] for the evaluation from a to b, we have

I[0, 2π][0,4+4sinθ] r dr dθ = I[0, 2π] 1/2r² E[0,4 + 4sinθ] dθ =

I[0, 2π] 16 + 32sinθ + 16sin²θ dθ substituting sin²θ = 1/2 - 1/2cos2θ, 16sin²θ = 8 - 8 cos2θ

I[0, 2π] 24 + 32sinθ - 8cos 2θ dθ =

24θ - 32cosθ - 4sin2θ e[0, 2π] = 24*2π = 48π