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 1126: 39

Answer

$2 \pi^2$

Work Step by Step

Here, we have $W=\int_C F\cdot dr$ $=\int_0^{2 \pi} [(t-\sin t) i+(3-\cos t)j) \cdot ((1-\cos t) i+\sin t j) dt$ $=\int_0^{2 \pi} t-\sin t-t \cos t+\cos t \sin t+3\sin t-\sin t \cos t dt$ $=\int_0^{2 \pi} t-t \cos t+2 \sin t dt$ $=\int_0^{2 \pi} -t \cos t+[\dfrac{t^2}{2}-2 \cos t ]_0^{2 \pi}$ $=[\int t \cos t dt]_0^{2 \pi}+2 \pi^2$ $W=\int_C F\cdot dr=[\int t \cos t dt]_0^{2 \pi}+2 \pi^2$ Solve $I=[\int t \cos t dt]_0^{2 \pi}=[ t \int\cos t dt-(\int \dfrac{dt}{dt} \int \cos t dt) dt]_0^{2 \pi}$ $=-0+2 \pi^2 $ $=2 \pi^2$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.