   Chapter 10, Problem 110RE

Chapter
Section
Textbook Problem

Finding the Arc Length of a Polar Curve In Exercises 109 and 110, find the length of the curve over the given interval.Polar Equation Interval r = 3 ( 1 − cos θ ) [ 0 , π ]

To determine

To calculate: The length of the curve given as, r=3(1cosθ) over the interval [0,π].

Explanation

Given:

The polar equation is;

r=3(1cosθ)

And interval is [0,π].

Formula used:

The length of the graph of r=f(θ) from θ=α to θ=β is given by;

s=αβr2+(drdθ)2dθ

The trigonometric identity, sin2θ+cos2θ=1 and sin2(θ2)=1cosθ2.

Calculation:

Consider the given polar equation,

r=3(1cosθ)

Now, find the derivative with respect to θ as;

drdθ=ddθ[3(1cosθ)]=ddθ(3)ddθ(3cosθ)=0(3sinθ)=3sinθ

Then, the length of the curve over the interval [0,π] is;

s=0πr2+(drdθ)2dθ=0π[3(1cosθ)]2+(3sinθ)2dθ=0π918cos

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