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05 1 00 2 0 area 0 5 1 t2 2t t3 4 6 1 0 11 12 1 1 2

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Unformatted text preview: y y (θ) = 2 sin θ sin θ = 2 sin2 (θ) and = 4 sin (θ) cos (θ). dθ √ dy dx |θ=π/6 = 1. and |θ=π/6 = 3. Therefore dθ √dθ 13 . , u= 22 √ 9. Determine the area between the polar curves r (t) = t and r (t) = 2 − t as shown. 0.5 1 0.0 2 0 Area= 0 5 1 = − t2 + 2t + t3 4 6 1 0 11 = 12 1 1 2 − 2t + t2 − t dt 2 2 10. When a bicycle wheel with radius 12 inches turns, the path that is taken by a spot on the tire is called a cycloid and its parametric equations are given as: x(t) = 12t − 12 sin t y (t) = 12 − 12 cos t 20 15 10 5 0 0 20 40 60 Determine the arc length of one arch of the cycloid. 2 dx = (12 − 12 cos (t))2 = 144 − 288 cos (t) + 144 cos2 (t) dt 2 dy = 144 sin2 (t) dt 2π Arc length= 0 288 − 288 cos (t) dt = 2π 0 12 2(1 − cos (t)) dt using the half-angle formula 2π = 0 2π 12 4 sin2 (t/2) dt = 0 24 sin (t/2) dt = −48 cos (t/2)|2π = 96 0...
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