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: 15



Work Step by Step

We know that the line equation can be defined as: $$ r_0+kt=(0,0,0)+t \space \lt 1,2,3 \gt=\lt t, 2t, -3t \gt \\ x=t \implies dx= dt \\ y=2 \space t \implies dy=2 \space dt \\z=3t \implies dz=3 \space dt $$ Substitute all the above values in the given integral as follows: $$ \int_{0,0,0}^{1,2,3} \space (4t^2)dt+2(t-9t^2) \space dt-(2)(2t)(3t) \space 3 \space dt=-16$$
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