Explanation Given data: x = a ( t − sin t ) (1) y = a ( 1 − cos t ) (2) 0 ≤ t ≤ 2 π . Calculation: Consider curve’s density to be δ = 1 . Find d x d t 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 − d d t sin t ) = a ( 1 − cos t ) Consider the expression for a typical segment of the curve with a mass ( d m ) . d m = d A = y d x = y ( d x d t ) d t Substitute a ( 1 − cos t ) for d x d t and a ( 1 − cos t ) for y as follows. d m = a ( 1 − cos t ) a ( 1 − cos t ) d t = a 2 ( 1 − cos t ) 2 d t Consider the curve’s mass for 0 ≤ t ≤ 2 π is, M = ∫ 0 2 π d m = ∫ 0 2 π a 2 ( 1 − cos t ) 2 d t = a 2 ∫ 0 2 π ( 1 − 2 cos t + cos 2 t ) d t = a 2 ∫ 0 2 π ( 1 − 2 cos t + 1 2 + 1 2 cos 2 t ) d t Simplify the equation as follows. M = a 2 [ 3 2 t − 2 sin t + sin 2 t 4 ] 0 2 π = a 2 [ 3 2 ( 2 π ) − 2 sin ( 2 π ) + sin 4 π 4 − 0 ] 0 2 π = 3 π a 2 Find moment about x -axis x ˜ . x ˜ = x = a ( t − sin t ) Find moment about y -axis y ˜ . y ˜ = 1 2 y = 1 2 a ( 1 − cos t ) Find M x . M x = ∫ y ˜ d m = ∫ y ˜ δ d A = ∫ 0 2 π 1 2 a ( 1 − cos t ) a 2 ( 1 − cos t ) 2 d t = a 3 2 ∫ 0 2 π ( 1 − cos t ) 3 d t Simplify the equation as follows

Thomas' Calculus Format: Unbound (saleable) With Access Card

14th Edition

ISBN: 9780134768762

Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.

Publisher: Prentice Hall

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Chapter 11, Problem 24AAE

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Find the coordinates of the centroid of the curve x=a(t−sint), y=a(1−cost), 0≤t≤2π.

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Chapter 11, Problem 23AAE

Chapter 11, Problem 25AAE

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Thomas' Calculus Format: Unbound (saleable) With Access Card

Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Finding Cartesian from Parametric...

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Find the coordinates of the centroid of the curve x = a ( t − sin t ) , y = a ( 1 − cos t ) , 0 ≤ t ≤ 2 π .

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Find the coordinates of the centroid of the curve x=a(t−sint), y=a(1−cost), 0≤t≤2π.

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