Explanation Given information: The given field is F = 〈 − 2 , y 〉 and the given curve is a half-circle x 2 + y 2 = 1 with y ≥ 0 oriented counter clockwise. Concept Used: The line integral of the given vector field F = f 1 i + f 2 j over the curve C is given by ∫ C F . d r = ∫ a b F ( r ( t ) ) ⋅ r ' ( t ) d t where r ( t ) is the parametrization of the given curve C. Calculation: As given F = 〈 − 2 , y 〉 , first we parameterize the given curve which is a half-circle x 2 + y 2 = 1 with y ≥ 0 oriented counter clockwise. r ( t ) = 〈 cos t , sin t 〉 ; 0 ≤ t ≤ π as y ≥ 0 so the half-circle is the upper half. Thus, F ( r ( t ) ) = 〈 − 2 , sin t 〉 r ' ( t ) = 〈 − sin t , cos t 〉 The product of F ( r ( t ) ) and r ' ( t ) is, F ( r ( t ) ) ⋅ r ' ( t ) = 2 sin t + sin t cos t = 2 sin t + 2 2 sin t cos t = 2 sin t + sin 2 t 2 Consider the integral, ∫ C F

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.2, Problem 20E

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To find:The line integral ∫CF⋅dr over the specified curve C.

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Chapter 17.2, Problem 19E

Chapter 17.2, Problem 21E

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Chapter 17 Solutions

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The line integral ∫ C F ⋅ d r over the specified curve C.

Solution Summary: The author calculates the line integral of the given vector field F=langle -2,yrangle over the specified curve C.

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To find:The line integral ∫CF⋅dr over the specified curve C.

Want to see the full answer?

Chapter 17 Solutions