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Chapter 11: Q. 28 (page 889)

In Exercises 28鈥�30 show that the given vector-valued functions are not arc length parametrizations. Then find an arc length parametrization for the curves defined by those functions.

r(t)=(3cost,3sint)

Short Answer

Expert verified

The arc length parametrization of the given curve isr(s)=3coss3,3sins3

Step by step solution

01

Step 1. Given information.

The given function isr(t)=鉄�3cost,3sint鉄�.

02

Step 2. Required answer.

Now,

r'(t)=(-3sint)2+(3cost)2=9sin2t+cos2t=3Consider鈭�cdr'(t)dt=鈭�cd3dt=3tcd=3(d-c)鈮�d-c

Hence the given vector-valued functions are not arc length parameterizations on the interval [0,1]the arc length of the curve is 3(1-0)=3

replace t withs3inr(t)Weget,r(s)=3coss3,3sins3

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