ConsidertheVector functiona.) Find thet:0arc kkng thof rle) fromtob.)Find the curvature a t=

The given vector function is rt=t,t2,23t3. a) Find the arc length of rt from t=0 to t=1 as…

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Consider the Vector function a.) Find the もこ0 arc leng th of rlt) from to b.) Find the curvature a t=l

Elementary Geometry for College Students

6th Edition

ISBN:9781285195698

Author:Daniel C. Alexander, Geralyn M. Koeberlein

Publisher:Daniel C. Alexander, Geralyn M. Koeberlein

Chapter10: Analytic Geometry

Section10.4: Analytic Proofs

Problem 30E: Complete an analytic proof of the following theorem: In a triangle that has sides of lengths a,b,...

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Question

Consider
the
Vector function
a.) Find the
t:0
arc kkng th
of rle) from
to
b.)
Find the curvature a t=

Expert Solution

Step 1

The given vector function is $r (t) = <t, t^{2}, \frac{2}{3} t^{3}>$ .

Find the arc length of $r (t)$ from $t = 0 to t = 1$ as follows.

The arc length of the curve null from $t = a to t = b$ is given by,

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

Here,

$x (t) = t y (t) = t^{2} z (t) = \frac{2}{3} t^{3}$

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