   Chapter 10.3, Problem 54E

Chapter
Section
Textbook Problem

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

To determine

To calculate: The arc length of curve x=cosθ+θsinθ,y=sinθθcosθ on the interval [0,2π].

Explanation

Given:

Parametric equations,

x=cosθ+θsinθy=sinθθcosθ

Formula used:

Arc length of curve is given by:

s=02π((dxdθ)2+(dydθ)2)dθ

Calculation:

Consider the given equations,

x=cosθ+θsinθy=sinθθcosθ

Differentiate x=cosθ+θsinθ with respect to t, to get,

dxdθ=sinθ+sinθ+θcosθ=θcosθ

Differentiate y=sinθθcosθ with respect to t, to get,

dydθ=cosθcosθ+θsinθ=θsinθ

Arc length of curve is given by:

s=02π((dxdθ)2+(

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