Final answer:
To find the area inside of r = 2cosθ and outside r = 1, you need to integrate the equation for the outer curve and subtract the integral of the equation for the inner curve. Set the two equations equal to each other to find the values of θ where the curves intersect. The area can be found using the limits of integration and the formula A = ∫(outer curve equation) - ∫(inner curve equation) dθ.
Step-by-step explanation:
To find the area inside of r = 2cosθ and outside r = 1, we need to determine the limits of integration for the polar coordinate system. The first equation, r = 2cosθ, represents a circle centered at the origin with a radius of 2. The second equation, r = 1, represents a circle centered at the origin with a radius of 1. To find the area between the two curves, we need to integrate the equation for the outer curve and subtract the integral of the equation for the inner curve.
The area can be found using the formula A = ∫(outer curve equation) - ∫(inner curve equation) dθ. In this case, the outer curve equation is r = 2cosθ and the inner curve equation is r = 1. We need to find the limits of integration, which are the values of θ where the curves intersect. By setting the two equations equal to each other, we can find the values of θ that satisfy the equation.
2cosθ = 1 ⟹ cosθ = 1/2 ⟹ θ = π/3 and 5π/3.
Therefore, the area inside of r = 2cosθ and outside r = 1 is given by A = ∫(2cosθ) dθ - ∫(1) dθ from θ = π/3 to θ = 5π/3.