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: 22

Answer

$0$

Work Step by Step

Here, we have $F(r(t)=\cos t i +\sin t j+\cos t \sin t k$ and $dr=(-\sin t i+\cos t j+k) dt$ $\int_{C} \overrightarrow{F} \cdot \overrightarrow{dr}=\int_0^{\pi} (\cos t i +\sin t j+\cos t \sin t k) \cdot (-\sin t i+\cos t j+k) d t$ $= \int_0^{\pi} (\cos t) (\sin t) dt$ $= \int_0^{\pi} (\dfrac{1}{2}) [2 (\cos t) (\sin t)] dt$ $=(1/2) \int_0^{\pi} \sin (2 t) dt$ $=(1/2)[\dfrac{-\cos (2t) }{2}]_0^{\pi}$ $=0$
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