Explanation Given data: Parametric equations are x = t − sin t and y = 1 − cos t . Consider a curve C 1 , 0 ≤ t ≤ 2 π as arch of the cycloid from ( 0 , 0 ) to ( 2 π , 0 ) with parametric equations x = t − sin t (1) y = 1 − cos t (2) Differentiate equation (1) with respect to t . d d t ( x ) = d d t ( t − sin t ) d x d t = d d t ( t ) − d d t ( sin t ) d x d t = 1 − cos t { ∵ d d t ( t ) = 1 , d d t ( sin t ) = cos t } d x = ( 1 − cos t ) d t Differentiate equation (2) with respect to t . d d t ( y ) = d d t ( 1 − cos t ) d y d t = d d t ( 1 ) − d d t ( cos t ) d y d t = 0 − ( − sin t ) { ∵ d d t ( k ) = 0 , d d t ( cos t ) = − sin t } d y = sin t d t Consider a curve C 2 , 0 ≤ t ≤ 2 π as a segment from ( 2 π , 0 ) to ( 0 , 0 ) with parametric equations x = 2 π − t (3) y = 0 (4) Differentiate equation (3) with respect to t . d d t ( x ) = d d t ( 2 π − t ) d x d t = d d t ( 2 π ) − d d t ( t ) d x d t = 0 − 1 { ∵ d d t ( t ) = 1 , d d t ( k ) = 0 } d x = − d t Differentiate equation (4) with respect to t . d d t ( y ) = d d t ( 0 ) d y d t = 0 { ∵ d d t ( 0 ) = 0 } d y = 0 d t The area of curve C , which is one arch of cycloid is the union of regions of curves C 1 and C 2

The area under one arch of the cycloid.

Question

Chapter 16.4, Problem 19E

To determine

To find: The area under one arch of the cycloid.

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Chapter 16.4, Problem 18E

Chapter 16.4, Problem 20E

Calculus: Early Transcendentals

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Calculus: Early Transcendentals

Vector Calculus. 19E

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Chapter 16 Solutions

Calculus: Early Transcendentals

Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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