Final answer:
To find the area of the region outside the circle r=3 and inside the circle r= -6 sinθ, calculate the area of the larger circle and subtract the area of the smaller circle.
Step-by-step explanation:
To find the area of the region outside the circle r=3 and inside the circle r= -6 sinθ, we can calculate the area of the larger circle with radius 3 and subtract the area of the smaller circle with radius -6 sinθ.
The formula for the area of a circle is A=πr². So, the area of the larger circle is A = π(3)² = 9π. The area of the smaller circle is A = π(-6 sinθ)² = 36π sin²θ.
Therefore, the area of the region is 9π - 36π sin²θ.