Chapter 10, Problem 110RE

### 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

# 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(1âˆ’cosÎ¸)

And interval is [0,Ï€].

Formula used:

The length of the graph of r=f(Î¸) from Î¸=Î± to Î¸=Î² is given by;

s=âˆ«Î±Î²r2+(drdÎ¸)2dÎ¸

The trigonometric identity, sin2Î¸+cos2Î¸=1 and sin2(Î¸2)=1âˆ’cosÎ¸2.

Calculation:

Consider the given polar equation,

r=3(1âˆ’cosÎ¸)

Now, find the derivative with respect to Î¸ as;

drdÎ¸=ddÎ¸[3(1âˆ’cosÎ¸)]=ddÎ¸(3)âˆ’ddÎ¸(3cosÎ¸)=0âˆ’(âˆ’3sinÎ¸)=3sinÎ¸

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

s=âˆ«0Ï€r2+(drdÎ¸)2dÎ¸=âˆ«0Ï€[3(1âˆ’cosÎ¸)]2+(3sinÎ¸)2dÎ¸=âˆ«0Ï€9âˆ’18cos

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