Explanation Formula Used: Green’s Theorem – I = ∮ C F ⋅ d r = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d A Given: F ( x , y ) = 〈 x 3 , 4 x 〉 Calculation: To compute integral of the given path we have to compute the area of the closed region shown in figure below using Green’s theorem, Using Green’s theorem and line integrals, we can say, I = ∮ C F ⋅ d r = ∬ D ( ∂ Q ∂ x − ∂ P ∂ y ) d A = ∫ C F ( x , y ) d r + ∫ B A ¯ F ( x , y ) d r ----------------------(1) Now let’s first find line integral along B A ¯ Therefore, we have to parameterize F ( x , y ) = 〈 x 3 , 4 x 〉 along B A ¯ x = − 1 and y = t − 1 ≤ t ≤ 0 Therefore, d x d t = 0 and d y d t = 1 Line integral of F along B A ¯ is given by, ∫ B A ¯ F ( x , y ) d r Now, F ( x , y ) = 〈 x 3 , 4 x 〉 = 〈 ( − 1 ) 3 , 4 ( − 1 ) 〉 = 〈 − 1 , − 4 〉 d r = 〈 d x d t , d y d t 〉 d t = 〈 0 , 1 〉 d t Therefore, ∫ B A ¯ F ( x , y ) d r = ∫ − 1 0 〈 − 1 , − 4 〉 ⋅ 〈 0 , 1 〉 d t ∫ B A ¯ F ( x , y ) d r = ∫ − 1 0 ( − 1 ) ( 0 ) + ( − 4 ) ( 1 ) d t ∫ B A ¯ F ( x , y ) d r = ∫ − 1 0 − 4 d t = − 4 ∫ − 1 0 d t ∫ B A ¯ F ( x , y ) d r = − 4 [ t ] − 1 0 = − 4 ( 0 − ( − 1 ) ) ∫ B A ¯ F ( x , y ) d r = − 4 --------------------------------(2) Now, using Green’s theorem let’s find area of the closed region D...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.1, Problem 14E

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To compute: integral of $F (x, y) = 〈 x^{3}, 4 x 〉$

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Chapter 18.1, Problem 13E

Chapter 18.1, Problem 15E

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Evaluate the line integrals using the Fundamental Theorem of Line Integrals: ∫c (yi+xj)*dr Where C is any path from (0,0) to (2,4).

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Chapter 18 Solutions

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Ch. 18.1 - Prob. 1PQ Ch. 18.1 - Prob. 2PQ Ch. 18.1 - Prob. 3PQ Ch. 18.1 - Prob. 4PQ Ch. 18.1 - Prob. 5PQ Ch. 18.1 - Prob. 1E Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E

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integral of F ( x , y ) = 〈 x 3 , 4 x 〉

Solution Summary: The author explains how to compute the integral of the given path using Green’s theorem and line integrals.

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To compute: integral of $F (x, y) = 〈 x^{3}, 4 x 〉$

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Chapter 18 Solutions

integral of F ( x , y ) = 〈 x 3 , 4 x 〉

Solution Summary: The author explains how to compute the integral of the given path using Green’s theorem and line integrals.

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To compute: integral of F(x,y)=〈x3,4x〉

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Chapter 18 Solutions

To compute: integral of $F (x, y) = 〈 x^{3}, 4 x 〉$