Q.106. Application of Integration, Area under the curve.

Question 106.
Find the area under the curve given by:
x = a(x - sin x), y = a(1 - cos x) in the interval x ∈ [0,2π].

Answer 106.
x = a(θ - sinθ)
=> dx = a(1 - cosθ) dθ
=> ydx
= [a(1 - cosθ] * [a(1 - cosθ] dθ
= a^2 * (1 - cosθ)^2 dθ
= (a^2/2) (2 - 4cosθ + 2cos^2 θ) dθ
= (a^2/2) (3 - 4cosθ + cos2θ) dθ

 => Required area
= (a^2/2) ∫ (0 to 2π) (3 - 4cosθ + cos2θ) dθ
= (a^2/2) [3θ - 4sinθ + (1/2)sin2θ] (θ=0 to 2π)
= (a^2/2) [3*2π]
= 3πa^2.

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