Chapter 10.3, Problem 57E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Arc Length In Exercises 55-58, find the arc length of the curve on the interval [ 0 , 2 π ] .Cycloid arch: x = a ( θ − sin θ ) , y = a ( 1 − cos θ )

To determine

To calculate: The arc length of curve x=a(θsinθ),y=a(1cosθ) on the interval [0,2π].

Explanation

Given:

Parametric equations,

x=a(Î¸âˆ’sinÎ¸)y=a(1âˆ’cosÎ¸)

Formula used:

Arc length of curve is given by:

s=âˆ«02Ï€((dxdÎ¸)2+(dydÎ¸)2)dÎ¸

And,

sin2Î¸2=1âˆ’cosÎ¸2sin2Î¸+cos2Î¸=1

Calculation:

Consider the given equations,

x=a(Î¸âˆ’sinÎ¸)y=a(1âˆ’cosÎ¸)

Differentiate x=a(Î¸âˆ’sinÎ¸) with respect to t, to get,

dxdÎ¸=a(1âˆ’cosÎ¸)

Differentiate y=a(1âˆ’cosÎ¸) with respect to t, to get,

dydÎ¸=asinÎ¸

Arc length of curve is given by:

s=âˆ«02Ï€((dxdÎ¸)2+(dydÎ¸)2)dÎ¸

Substitute the values of dxdt and dy

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