Explanation Given information: A curve C is specified by C : Circleof radius R , centered at originoriented counterclockwise A vector valued function F = 〈 − y ( x 2 + y 2 ) 2 , x ( x 2 + y 2 ) 2 〉 Formula used: Suppose that curve C is a parametric curve r ( t ) where a ≤ t ≤ b , then the line integral of ∫ C F ⋅ d r is given by ∫ C F ⋅ d r = ∫ C F ( r ( t ) ) ⋅ r ′ ( t ) d t Calculation: Consider the curve C. Circleof radius R , centered at origin oriented counterclockwise Let us consider the parametric equation for the portion. x = R cos t and y = R sin t , such that 0 ≤ t ≤ 2 π C : r ( t ) = 〈 R cos t , R sin t 〉 So, r ′ ( t ) = 〈 − R sin t , R cos t 〉 And F ( r ( t ) ) = F ( R cos t , R sin t ) = 〈 − R sin t ( R 2 cos 2 t + R 2 sin 2 t ) 2 , R cos t ( R 2 cos 2 t + R 2 sin 2 t ) 2 〉 = 〈 − R sin t R 4 , R cos t R 4 〉 The value of the line integral can be calculated as shown below

4th Edition

ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.2, Problem 24E

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To evaluate:

The integral ∫CF⋅dr for the oriented curve specified.

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Chapter 16.2, Problem 23E

Chapter 16.2, Problem 25E

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Chapter 16 Solutions

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E

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The integral ∫ C F ⋅ d r for the oriented curve specified.

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To evaluate: The integral ∫CF⋅dr for the oriented curve specified.

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Chapter 16 Solutions

To evaluate:

The integral ∫CF⋅dr for the oriented curve specified.