(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

See similar textbooks

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16.3, Problem 12E

Chapter 16.3, Problem 14E

Students have asked these similar questions

A graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

The graph below is the function f(z) 4 3 -2 -1 -1 1 2 3 -3 Consider the function f whose graph is given above. (A) Find the following. If a function value is undefined, enter "undefined". If a limit does not exist, enter "DNE". If a limit can be represented by -∞o or ∞o, then do so. lim f(z) +3 lim f(z) 1-1 lim f(z) f(1) = 2 = -4 = undefined lim f(z) 1 2-1 lim f(z): 2-1+ lim f(x) 2+1 -00 = -2 = DNE f(-1) = -2 lim f(z) = -2 1-4 lim f(z) 2-4° 00 f'(0) f'(2) = = (B) List the value(s) of x for which f(x) is discontinuous. Then list the value(s) of x for which f(x) is left- continuous or right-continuous. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5). If there are none, enter "none". Discontinuous at z = Left-continuous at x = Invalid use of a comma.syntax incomplete. Right-continuous at z = Invalid use of a comma.syntax incomplete. (C) List the value(s) of x for which f(x) is non-differentiable. Enter your answer as a comma-separated list, if needed (eg. -2, 3, 5).…

A graph of the function f is given below: Study the graph of f at the value given below. Select each of the following that applies for the value a = -4. f is defined at = a. f is not defined at 2 = a. If is continuous at x = a. Of is discontinuous at x = a. Of is smooth at x = a. f is not smooth at x = a. If has a horizontal tangent line at x = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. Of has no tangent line at x = a. f(a + h) − f(a) h lim is finite. h→0 f(a + h) - f(a) lim is infinite. h→0 h f(a + h) - f(a) lim does not exist. h→0 h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

Chapter 16 Solutions

Calculus: Early Transcendentals

Knowledge Booster

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