The area of the region is approximately 47.1239 square units and the arc length of the polar curve is 25π units.
Area of the Region Inside r = 10sinθ and Outside r = 3
Set the equations equal to find where they intersect: 10sinθ = 3
Solve for θ: θ = arcsin(3/10) ≈ 0.3044 radians and θ = π - 0.3044 ≈ 2.8379 radians
The region lies between θ = 0.3044 and θ = 2.8379.
Area = 1/2 ∫[0.3044, 2.8379] [(10sinθ)^2 - 3^2] dθ
Area ≈ 47.1239 square units (rounded to four decimal places)
Arc Length of the Polar Curve r = 5 + 5cosθ
Formula for Arc Length in Polar Coordinates:
L = ∫[a, b] √[r^2 + (dr/dθ)^2] dθ
dr/dθ = -5sinθ
L = ∫[0, 2π] √[(5 + 5cosθ)^2 + (-5sinθ)^2] dθ
L ≈ 25π units (exact value)