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Find the area of the region inside r = 10sin theta and outside r = 3 , round your answer to four digits after the decimal. QUESTION 5 Find the arc length of the polar curve described by r = 5 + 5cos theta

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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)

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