I need help solving this problem, I think I\'m messing it up greatly (My work st

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I need help solving this problem, I think I'm messing it up greatly (My work starts with "If you have a function..." Also I need to know how to plot this using maple.

I will add points for an extra detailed answer.

When recording live performances sound engineers often use a microphone with a cardiod pickup pattern because it suppresses noise from the audience. Suppose the microphone is place 4 meters from the front of a stage, facing the stage, and the boundary of the microphones recording area is the cardiod r = 8 + 8 sin(theta). where r is measured in meters and the microphone Is at the pole. The musicians need to know where on stage the optima! pickup range of the microphone so they arc recorded clearly. If you have a function r=f(theta). then the area between the origin and the curve between a le theta le b is 1/2 (f(theta))2 d theta 1/2 (8 + 8 sin(theta))2 d theta 32 (1 + sin(theta))2 d theta 32 1 + 2 sin(theta) + sin 2 (theta) d theta Since cos(2x) - cos (x2) - sin (x2) - 1 - 2 sin (x)2, then sin (theta)2 = (cos(2 theta) - 1)/2 32 1 + 2 sin(theta) + (cos(2 theta) - 1)/2 d theta 32 [theta - 2 cos (theta) + (1/4) sin (2 theta) - (1/2) theta]

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