(a.) Explanation Given: The vector field, F x , y = 3 x 2 + y 2 i + 2 x y j and the curve C as shown below: Concept used: The value of an integration can be calculated directly by substituting the value of the vector field and the curve directly. Calculation: The given vector field is F x , y = 3 x 2 + y 2 i + 2 x y j and the given curve C is given as y = − 4 − x 2 . Differentiating, d y = − 1 2 4 − x 2 − 2 x d x Simplifying, d y = x 4 − x 2 d x Now, the curve can be given as r x , y = x i + y j , where y = − 4 − x 2 from x = − 2 to x = 2 . Differentiating, d r x , y = d x i + d y j Put F x , y = 3 x 2 + y 2 i + 2 x y j and d r x , y = d x i + d y j in ∫ C F ⋅ d r to get, ∫ C F ⋅ d r = ∫ C 3 x 2 + y 2 i + 2 x y j ⋅ d x (b.) To determine To Show : F is conservative and to find a function f such that F = ∇ f . (c.) To determine The value of ∫ C F ⋅ d r using the Fundamental Theorem of Calculus. (d.) To determine The value of ∫ C F ⋅ d r by replacing C by a simpler curve with the same end points.

9th Edition

ISBN: 9780357598511

Author: James Stewart; Daniel K. Clegg; Saleem Watson

Publisher: Cengage Learning US

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Question

Chapter 16.3, Problem 13E

(a.)

To determine

The value of ∫CF⋅dr directly.

(b.)

To determine

To Show: F is conservative and to find a function f such that F=∇f .

(c.)

To determine

The value of ∫CF⋅dr using the Fundamental Theorem of Calculus.

(d.)

To determine

The value of ∫CF⋅dr by replacing C by a simpler curve with the same end points.

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Chapter 16.3, Problem 12E

Chapter 16.3, Problem 14E

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Chapter 16 Solutions

Calculus: Early Transcendentals

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The value of ∫ C F ⋅ d r directly.

The value of ∫CF⋅dr directly.

To Show: F is conservative and to find a function f such that F=∇f .

The value of ∫CF⋅dr using the Fundamental Theorem of Calculus.

The value of ∫CF⋅dr by replacing C by a simpler curve with the same end points.

Want to see the full answer?

Chapter 16 Solutions