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Find the arc length function for curve: r(t) = (cos(-2t), -2t, sin(-2t)) starting from r(-5). Use...

Question:

Find the arc length function for curve: {eq}r(t) = \langle \cos(-2t), \; -2t, \; \sin(-2t) \rangle {/eq} starting from {eq}r(-5) {/eq}. Use your arc length function to quick evaluate the length of {eq}r {/eq} from:

{eq}t = -5 {/eq} to {eq}t = -3 {/eq} and

{eq}t = -5 {/eq} to {eq}t = 2 {/eq} .

Arc Length:

If {eq}r(t)= \left\langle x(t), y(t), z(t) \right\rangle {/eq} is the parametric equation of a curve, then the length of this curve between the t values, t=a and t=b is given by,

Arc length =$$L=\int_{a}^{b}\sqrt {(x'(t)^{2}+y'(t)^{2}+z'(t)^{2}} dt $$

Answer and Explanation: 1

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The given curve is:

$$r(t) = \left\langle \cos (-2 t),-2 t, \sin (-2 t) \right\rangle= \left\langle x(t), y(t), z(t) \right\rangle $$

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How to Find the Arc Length of a Function

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Chapter 12 / Lesson 12
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You don't always walk in a straight line. Sometimes, you want to know the distance between two points when the path is curved. In this lesson, you'll learn about finding the length of a curve.


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