The parametric equation of a cycloid is given below.
What is the area of the region bounded by the two arcs of the cycloid in the above figure?
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The area of the curve can be written as A = ∫ x = x 0 x = x 1 y d x = ∫ x = x 0 x = x 1 y d t d x d t
Where the parameters of initial and final points correspond to t = 0 and t = 4 π . Hence A = ∫ 0 4 π a 2 ( 1 − cos t ) 2 d t = 6 π a 2