Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 17 - Line and Surface Integrals - 17.2 Line Integrals - Exercises - Page 933: 42

Answer

Figure A: the line integral of the vector field is positive Figure B: the line integral of the vector field is positive Figure C: the line integral of the vector field is positive

Work Step by Step

Figure A: the line integral of the vector field is positive Explanation: the dot products ${\bf{F}}\cdot{\bf{T}}$ are positive along the path because the angle between the vectors are acute. Figure B: the line integral of the vector field is positive Explanation: most of the dot products ${\bf{F}}\cdot{\bf{T}}$ are positive along the path because the angle between the vectors are obtuse. Therefore, the line integral is positive. Figure C: the line integral of the vector field is positive Explanation: most of the dot products ${\bf{F}}\cdot{\bf{T}}$ are positive along the path because the angle between the vectors are acute. Though, there are some negative dot products closer to the end of the path, the positive values are dominant. Therefore, the line integral is positive.
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