   Chapter 10.3, Problem 53E

Chapter
Section
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(θsinθ)y=a(1cosθ)

Formula used:

Arc length of curve is given by:

s=02π((dxdθ)2+(dydθ)2)dθ

And,

sin2θ2=1cosθ2sin2θ+cos2θ=1

Calculation:

Consider the given equations,

x=a(θsinθ)y=a(1cosθ)

Differentiate x=a(θsinθ) with respect to t, to get,

dxdθ=a(1cosθ)

Differentiate y=a(1cosθ) with respect to t, to get,

dydθ=asinθ

Arc length of curve is given by:

s=02π((dxdθ)2+(dydθ)2)dθ

Substitute the values of dxdt and dy

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