Use a graphing utility to graph the polar equations. Find the area of the given region analytically: between the loops of r = 4-8 cos(θ)
Answer (2)
Find the angles where r = 0 : cos ( θ ) = 2 1 , so θ = 3 π and θ = 3 5 π .
Set up the integral for the area between the loops: A = 2 1 ∫ 3 π 3 5 π ( 4 − 8 cos ( θ ) ) 2 d θ .
Evaluate the integral using trigonometric identities: A = 32 π + 32 3 .
The area between the loops is 32 π + 32 3 .
Explanation
Problem Setup We are asked to find the area between the inner and outer loops of the polar equation r = 4 − 8 cos ( θ ) .
Finding Integration Limits First, we need to find the angles where r = 0 to determine the limits of integration. Setting 4 − 8 cos ( θ ) = 0 , we get cos ( θ ) = 2 1 . The solutions are θ = 3 π and θ = 3 5 π .
Setting up the Integral The area between the loops is given by the integral
A = 2 1 ∫ 3 π 3 5 π ( 4 − 8 cos ( θ ) ) 2 d θ
Expanding the integrand, we have
A = 2 1 ∫ 3 π 3 5 π ( 16 − 64 cos ( θ ) + 64 cos 2 ( θ )) d θ
Using the identity cos 2 ( θ ) = 2 1 + c o s ( 2 θ ) , we get
A = 2 1 ∫ 3 π 3 5 π ( 16 − 64 cos ( θ ) + 32 + 32 cos ( 2 θ )) d θ
A = 2 1 ∫ 3 π 3 5 π ( 48 − 64 cos ( θ ) + 32 cos ( 2 θ )) d θ
Evaluating the Integral Now, we integrate:
A = 2 1 [ 48 θ − 64 sin ( θ ) + 16 sin ( 2 θ ) ] 3 π 3 5 π
A = 2 1 [ ( 48 ⋅ 3 5 π − 64 ⋅ sin ( 3 5 π ) + 16 ⋅ sin ( 3 10 π ) ) − ( 48 ⋅ 3 π − 64 ⋅ sin ( 3 π ) + 16 ⋅ sin ( 3 2 π ) ) ]
A = 2 1 [ ( 80 π − 64 ⋅ ( − 2 3 ) + 16 ⋅ ( 2 3 ) ) − ( 16 π − 64 ⋅ ( 2 3 ) + 16 ⋅ ( 2 3 ) ) ]
A = 2 1 [ 80 π + 32 3 + 8 3 − 16 π + 32 3 − 8 3 ]
A = 2 1 [ 64 π + 64 3 ]
A = 32 π + 32 3
Final Answer Therefore, the area between the loops is 32 π + 32 3 ≈ 142.100 .
Examples
Understanding how to calculate the area between loops in polar equations is useful in various fields, such as designing antennas with specific radiation patterns or analyzing the trajectories of objects in orbital mechanics. For instance, engineers might use this concept to optimize the coverage area of a wireless signal by adjusting the shape of the antenna. Similarly, astronomers could apply this to study the regions of space affected by gravitational forces.
To find the area between the loops of the polar curve r = 4 − 8 cos ( θ ) , we identify intersection points where r = 0 , set up the area integral, and evaluate it using trigonometric identities. The area is calculated as 32 π + 32 3 .
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