   Chapter 10.5, Problem 72E

Chapter
Section
Textbook Problem

# HOW DO YOU SEE IT? Which graph, traced out only once, has a larger arc length? Explain your reasoning. To determine

To Calculate: Which of the given graph is traced out only once and has larger arc length.

Explanation

Given:

Graphs are given which are shown as below:

Formula Used:

L=0πf(θ)2+f'(θ)2

Calculation:

Equation of rose curve is given by: r=2cos3θ

In the above graph θ increases from 0 to π

L=0πf(θ)2+f'(θ)2=0π(2cos3θ)2+(6sin3θ)2=0π(4cos23θ)+(36sin23θ)=0π(1+(32sin23θ)

Further solve and get,

L=0π(16cos6θ+(16).dθ

Put cos6θ+1=t2.

Now,

=4202tdt6sin6θ=4202t2dt6(2t2t4)=402dt3(2t2)=4302dt((2)2t2)

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