Find the area inside the circle and outside the cardioid .
A) 7
E) 75.36

Question

Find the area inside the circle  and outside the cardioid  .
A) 7 
E) 75.36

Quizplus · Accepted Answer

The circle has radius 4, so its area is 16π.

The cardioid can be described by the polar equation

r = 2(1 + cos θ)

To find the points of intersection, we need to solve the equation

r = 4

2(1 + cos θ) = 4

1 + cos θ = 2

cos θ = 1

θ = 0, 2π

So the cardioid intersects the circle at the points (4,0) and (-4,0).

The area inside the circle and outside the cardioid is the area of the circle minus the area of the cardioid.

To find the area of the cardioid, we integrate the polar equation from θ = 0 to θ = 2π.

A = (1/2) ∫[0,2π] (r^2 - 2r*cos θ) dθ

= (1/2) ∫[0,2π] (4 - 4cos θ - 4cos^2 θ) dθ

= (1/2) ∫[0,2π] (6 - 4cos θ - 2cos 2θ) dθ

= (1/2) [6θ - 4sin θ - sin 2θ]∣∣∣0^{2π}

= 6π

So the area inside the circle and outside the cardioid is

16π - 6π = 10π

≈ 31.42

Which is closest to C: 11ead0bc_8e6d_c974_99a0_fb53d4b42495_TB6988_11.

Find the Area Inside the Circle and Outside the Cardioid