Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.2 Line Integrals - 16.2 Exercises - Page 1125: 31


$\dfrac{172,704}{5,632,705}\sqrt 2(1-e^{-14\pi})$

Work Step by Step

Here, we have $ds=\sqrt{(dx/dt)^2+(dy/dt)^2+(dz/dt)^2}$ This implies that $ds=\sqrt{(-e^{-t}\cos 4t-4e^{-t} \sin 4t)^2+(-e^{-t}\sin 4t-4e^{-t} \cos 4t)^2+(-e^{-t})^2}dt=e^{-t} \sqrt {18} dt$ Now, we have $\int_{C} x^2y^2zds=\int_{0}^{2 \pi} (-e^{-t}\cos 4t)^3 \cdot (e^{-t} \sqrt {18} dt)$ When we use calculator, then we have $\int_{C} x^2y^2zds=\dfrac{172,704}{5,632,705}\sqrt 2(1-e^{-14\pi})$
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