Okay, let's solve this step-by-step:
1) The equations for the two curves are:
r = sin(θ) and r = cos(θ)
2) We need to find the intersection points of these two curves. This is done by setting them equal and solving for θ:
sin(θ) = cos(θ)
=> θ = π/4
3) The intersection points are (1, π/4) and (1, 3π/4). The region lies between θ = π/4 and θ = 3π/4.
4) To find the area, we use the formula:
A = ∫θ=3π/4 θ=π/4 2πr dθ
5) Substitute r = sin(θ) or r = cos(θ):
A = ∫θ=3π/4 θ=π/4 2πsin(θ) dθ
= 2π ∫θ=3π/4 θ=π/4 sin(θ) dθ
6) Integrate:
A = 2π(cos(θ) - sin(θ) )|π/4 to 3π/4
= 2π(0 - 1) = 2π
7) Therefore, the area of the region is 2π square units.
Let me know if you have any other questions!