Final answer:
To find the area of the region outside the circle r=3 and inside the circle r=-6cosθ, we need to find the area of the larger circle and subtract the area of the smaller circle. The area of the larger circle is 9π and the area of the smaller circle is 3π. Therefore, the area of the region is 6π.
Step-by-step explanation:
To find the area of the region outside the circle r=3 and inside the circle r=-6cosθ, we need to find the area of the larger circle and subtract the area of the smaller circle.
The radius of the larger circle is 3, so its area is π(3)² = 9π. The radius of the smaller circle is given by r = -6cosθ.
To find its area, we need to solve for θ when r = -6cosθ = 0, which gives us θ = π/2 and 3π/2.
Since the smaller circle lies inside the larger circle, we need to find the area enclosed by the larger circle between these two values of θ.
This can be done by finding the arc length between the two points and multiplying it by the radius of the larger circle. In this case, the radius is 3 and the arc length is 3π/2 - π/2 = π.
Therefore, the area of the smaller circle is 3π. Finally, the area of the region is the area of the larger circle minus the area of the smaller circle: 9π - 3π = 6π.