Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 13: Vector-Valued Functions and Motion in Space - Practice Exercises - Page 777: 15


$\dfrac{\pi}{4}\sqrt {1+\dfrac{\pi^2}{16}}+\ln |\dfrac{\pi}{4}+\sqrt {1+\dfrac{\pi^2}{16}}|$

Work Step by Step

$v(t)=\dfrac{dr}{dt}=-2 \sin t i +2 \cos t j +2 t k$ and $|v(t)|=\sqrt {(-2 \sin t)^2 +(2 \cos t)^2 +(2 t)^2}=2 \sqrt {1+t^2}$ $Arc \ Length ( L)=\int_a^b 2 \sqrt {1+t^2} \ dt =[t \sqrt {1+t^2}+\ln |t + \sqrt {1+t^2}|]_0^{\pi/4}$ and $L=\dfrac{\pi}{4}\sqrt {1+\dfrac{\pi^2}{16}}+\ln |\dfrac{\pi}{4}+\sqrt {1+\dfrac{\pi^2}{16}}|$
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