Thomas' Calculus 13th Edition

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

Chapter 16: Integrals and Vector Fields - Section 16.3 - Path Independence, Conservative Fields, and Potential Functions - Exercises 16.3 - Page 967: 16


$$-\pi $$

Work Step by Step

We know that the line equation can be defined as: $$ r(t)=r_0+kt=(0,0,0)+t \space \lt 3,3,1 \gt=\lt 3t, 3t, t \gt \\ x=3t \implies dx= 3 dt \\ y=3 \space t \implies dy=3 dt \\ z=2t \implies dz=2 \space dt $$ Substitute all the above values in the given integral as follows: $$ \int_{0,0,0}^{3,3,1}18t \space dt-27 \space t^2 \space dt+\dfrac{-4}{1+t^2} dt=-\pi $$
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