To determine Find the area under one arch cycloid x = a ( t − sin t ) , y = a ( 1 − cos t ) . Explanation Given data: x = a ( t − sin t ) (1) y = a ( 1 − cos t ) Calculation: Find d x Differentiate with respect to t of equation (1) as follows. d x d t = d d t [ a ( t − sin t ) ] = a [ d d t ( t − sin t ) ] = a ( d d t t − d d t sin t ) = a ( 1 − cos t ) Rearrange the equation as follows. d x = a ( 1 − cos t ) d t Consider the general expression for area under cycloid. A = ∫ 0 2 π y d x Substitute a ( 1 − cos t ) for y and a ( 1 − cos t ) d t for d x as follows. A = ∫ 0 2 π a ( 1 − cos t ) a ( 1 − cos t ) d t = a 2 ∫ 0 2 π ( 1 − cos t ) ( 1 − cos t ) d t = a 2 ∫ 0 2 π ( 1 − cos t − cos t + cos 2 t ) d t = a 2 ∫ 0 2 π ( 1 − 2 cos t + cos 2 t ) d t Simplify the equation as follows

4th Edition

ISBN: 9780134439099

Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.

Publisher: Pearson,

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Chapter 11.2, Problem 21E

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Find the area under one arch cycloid x=a(t−sint),y=a(1−cost).

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Chapter 11.2, Problem 20E

Chapter 11.2, Problem 22E

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Thomas' Calculus

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