Question

When recording live performances, sound engineers often use a microphone with a cardioids pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 5 m from the front on the stage and the boundary of the optimal pickup region is given by the given cardioids, 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. The answer should be in two decimal places. r = 10 + 10sin(θ)

Answer 1

r = 4 + 4sinθ
y = rsinθ > 3 when π/6 < θ < 5π/6

So to find the shaded area we need to integrate
dA = rdr dθ
between the limits
r = 3/sinθ to 4 + 4sinθ
and θ = π/6 to 5π/6

You need to integrate r first, and you'll get
∫ rdrdθ = ∫ [r^2 /2] dθ
= ∫ [(4+4sinθ)^2 - (3cscθ)^2] dθ
= ∫ [16 + 32sinθ + 16sin^2 θ - 9csc^2 θ] dθ
= [24θ - 32cosθ + 9cotθ - 4sin(2θ)]

Apply the limits to get
16π + 18√3