Final answer:
To find the area of the region that lies inside the first curve and outside the second curve, you can set up an integral and solve for the area between the two curves. The integral can be simplified by splitting it into three parts and solving each part separately. The final result will give you the area of the region.
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 calculate the area between the two curves. Since the first curve is given by the equation r = 3 - 3sin(θ) and the second curve is given by the equation r = 3, we can set up the integral as:
A = ∫[0, 2π]((3 - 3sin(θ))^2 - 3^2)dθ
Now we can expand and simplify the integral:
A = ∫[0, 2π](9 - 18sin(θ) + 9sin^2(θ) - 9)dθ
The integral can be split into three parts:
A = ∫[0, 2π]9dθ - 18∫[0, 2π]sin(θ)dθ + 9∫[0, 2π]sin^2(θ)dθ - 9∫[0, 2π]dθ
Solving each integral gives:
A = 9(2π - 0) - 18(-cos(2π) + cos(0)) + 9(π - 0) - 9(2π - 0)