Explanation Formula Used: Area enclosed by C = ∮ C x d y = ∮ C y d x = 1 2 ∫ C x d y − y d x Given: The cycloid is parameterized by, r ( t ) = 〈 t − sin t , 1 − cos t 〉 for 0 ≤ t ≤ 2 π Calculation: We have been given, r ( t ) = 〈 t − sin t , 1 − cos t 〉 for 0 ≤ t ≤ 2 π To calculate the area of the given region, we have to find the line integrals of line segment O A ¯ and the cycloid Γ A = 1 2 ∫ O A ¯ x d y − y d x + 1 2 ∫ Γ x d y − y d x --------------(1) Now we have to parameterize along O A ¯ x = t 2 π ≥ t ≥ 0 and y = 0 Therefore, d x = d t and d y = 0 ∫ O A ¯ x d y − y d x = ∫ 0 2 π t ( 0 ) − 0 ( d t ) = 0 -------------------------(2) Now let’s parameterize the cycloid Γ x = t − sin t and y = 1 − cos t Therefore, d x = ( 1 − cos t ) d t and d y = sin t d t x d y − y d x = ( t − sin t ) sin t d t − ( 1 − cos t ) ( 1 − cos t ) d t x d y − y d x = ( t sin t − sin 2 t ) d t − ( 1 − 2 cos t + cos 2 t ) d t x d y − y d x = ( t sin t − sin 2 t − 1 + 2 cos t − cos 2 t ) d t x d y − y d x = ( t sin t + 2 cos t − 1 − sin 2 t − cos 2 t ) d t x d y − y d x = ( t sin t + 2 cos t − 1 − sin 2 t − cos 2 t ) d t x d y − y d x = ( t sin t + 2 cos t − 1 − ( sin ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 18.1, Problem 16E

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To calculate: the area of the region between the cycloid and the x-axis.

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Chapter 18.1, Problem 15E

Chapter 18.1, Problem 17E

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Chapter 18 Solutions

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