Final answer:
To find the area of the region that lies inside the first curve and outside the second curve, we need to determine the points of intersection between the two curves. The area is given by the integral of (14 cos θ - 7)^2 dθ from π/3 to 5π/3. Evaluating this integral, we find that the area is 154π square units.
Step-by-step explanation:
To find the area of the region that lies inside the first curve and outside the second curve, we need to determine the points of intersection between the two curves. The first curve is represented by the equation r = 14 cos θ and the second curve is represented by the equation r = 7. Setting the two equations equal to each other, we have 14 cos θ = 7. Solving for θ, we find that θ = π/3 and θ = 5π/3. These are the points of intersection between the two curves.
To find the area of the region between the curves, we need to integrate the difference between the first curve and the second curve from θ = π/3 to θ = 5π/3. The area is given by the integral of (14 cos θ - 7)^2 dθ from π/3 to 5π/3. Evaluating this integral, we find that the area is 154π square units.