Final answer:
To find the area of the region inside the polar curve r=8 cos θ and outside the curve r=5-2 cos θ, set the two equations equal to each other, find the points of intersection, and integrate the difference between the curves using polar coordinates.
Step-by-step explanation:
To find the area of the region inside the polar curve r=8 cos θ and outside the curve r=5-2 cos θ, we need to determine the points of intersection of the two curves. Set the two equations equal to each other: 8 cos θ = 5 - 2 cos θ. Simplifying, we get 10 cos θ = 5, or cos θ = 1/2. Solving for θ, we find two values: θ = π/3 and θ = 5π/3.
Next, we need to integrate the difference between the curves to find the area. In polar coordinates, the area element is given by dA = 1/2 (r1^2 - r2^2) dθ, where r1 is the outer curve and r2 is the inner curve. Using r1 = 8 cos θ and r2 = 5 - 2 cos θ, we can set up the integral:
∫[θ1,θ2] 1/2 ( (8 cos θ)^2 - (5 - 2 cos θ)^2 ) dθ
Substituting in the values of θ1 and θ2, the limits of integration, we can evaluate the integral to find the area.