Explanation Given: 1) F = 〈 − x 2 y , x , 0 〉 2) C is the close path A B C D 3) Need to evaluate ∫ C F . d r when C is the boundary of the rectangle with vertices A , B , C and D 5) Given fig for reference is as follows Formula used: 1) If A = ( x 1 , y 1 , z 1 ) and B = ( x 2 , y 2 , z 2 ) then equation of line A B is x − x 1 x 2 − x 1 = y − y 1 y 2 − y 1 = z − z 1 z 2 − z 1 2) ( a − b ) 2 = a 2 − 2 a b + b 2 3) ∫ x n d x = x n + 1 n + 1 Calculation: F = 〈 − x 2 y , x , 0 〉 [given] A = ( 6 , 0 , 4 ) , B = ( 6 , 3 , 0 ) , C = ( 0 , 3 , 0 ) , D = ( 0 , 0 , 4 ) [given vertices of rectangle A B C D ] Let’s calculate equation of line A B , B C , C D and D A which will be use in further calculation Equation of line A B x − 6 6 − 6 = y − 0 3 − 0 = z − 4 0 − 4 = t [using formula x − x 1 x 2 − x 1 = y − y 1 y 2 − y 1 = z − z 1 z 2 − z 1 ] x = 6 , y = 3 t , z = − 4 t − 4 d x = 0 , d y = 3 d t , d z = − 4 d t When y = 0 then t = 0 and when y = 3 then t = 1 So limit of t is 0 to 1 Equation of line B C x − 6 0 − 6 = y − 3 3 − 3 = z − 0 0 − 0 = t [using formula x − x 1 x 2 − x 1 = y − y 1 y 2 − y 1 = z − z 1 z 2 − z 1 ] x = − 6 t + 6 , y = 3 , z = 0 d x = − 6 d t , d y = 0 , d z = 0 When x = 6 then t = 0 and when x = 0 then t = 1 So limit of t is 0 to 1 Equation of line C D x − 0 0 − 0 = y − 3 0 − 3 = z − 0 4 − 0 = t [using formula x − x 1 x 2 − x 1 = y − y 1 y 2 − y 1 = z − z 1 z 2 − z 1 ] x = 0 , y = − 3 t + 3 , z = 4 t d x = 0 , d y = − 3 d t , d z = 4 d t When y = 3 then t = 0 and when y = 0 then t = 1 So limit of t is 0 to 1 Equation of line D A x − 0 6 − 0 = y − 0 0 − 0 = z − 4 4 − 4 = t [using formula x − x 1 x 2 − x 1 = y − y 1 y 2 − y 1 = z − z 1 z 2 − z 1 ] x = 6 t , y = 0 , z = 4 d x = 6 d t , d y = 0 , d z = 0 When x = 0 then t = 0 and when x = 6 then t = 1 So limit of t is 0 to 1 Calculating ∫ C F ... To determine b) Evaluate ∫ C F . d r using stoke’s theorem when C is the boundary of the wedge shaped box.

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Chapter 18.2, Problem 24E

To determine

(a)

Evaluate $\int_{C} F . d r$ considering $C$ is the boundary of the rectangle.

To determine

b)

Evaluate $\int_{C} F . d r$ using stoke’s theorem when $C$ is the boundary of the wedge shaped box.

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Chapter 18.2, Problem 23E

Chapter 18.2, Problem 25E

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Chapter 18 Solutions

CALCULUS (CLOTH)

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

Ch. 18.1 - Prob. 6E Ch. 18.1 - Prob. 7E Ch. 18.1 - Prob. 8E Ch. 18.1 - Prob. 9E Ch. 18.1 - Prob. 10E Ch. 18.1 - Prob. 11E Ch. 18.1 - Prob. 12E Ch. 18.1 - Prob. 13E Ch. 18.1 - Prob. 14E Ch. 18.1 - Prob. 15E Ch. 18.1 - Prob. 16E Ch. 18.1 - Prob. 17E Ch. 18.1 - Prob. 18E Ch. 18.1 - Prob. 19E Ch. 18.1 - Prob. 20E Ch. 18.1 - Prob. 21E Ch. 18.1 - Prob. 22E Ch. 18.1 - Prob. 23E Ch. 18.1 - Prob. 24E Ch. 18.1 - Prob. 25E Ch. 18.1 - Prob. 26E Ch. 18.1 - Prob. 27E Ch. 18.1 - Prob. 28E Ch. 18.1 - Prob. 29E Ch. 18.1 - Prob. 30E Ch. 18.1 - Prob. 31E Ch. 18.1 - Prob. 32E Ch. 18.1 - Prob. 33E Ch. 18.1 - Prob. 34E Ch. 18.1 - Prob. 35E Ch. 18.1 - Prob. 36E Ch. 18.1 - Prob. 37E Ch. 18.1 - Prob. 38E Ch. 18.1 - Prob. 39E Ch. 18.1 - Prob. 40E Ch. 18.1 - Prob. 41E Ch. 18.1 - Prob. 42E Ch. 18.1 - Prob. 43E Ch. 18.1 - Prob. 44E Ch. 18.1 - Prob. 45E Ch. 18.1 - Prob. 46E Ch. 18.1 - Prob. 47E Ch. 18.1 - Prob. 48E Ch. 18.1 - Prob. 49E Ch. 18.1 - Prob. 50E Ch. 18.1 - Prob. 51E Ch. 18.2 - Prob. 1PQ Ch. 18.2 - Prob. 2PQ Ch. 18.2 - Prob. 3PQ Ch. 18.2 - Prob. 4PQ Ch. 18.2 - Prob. 5PQ Ch. 18.2 - Prob. 1E Ch. 18.2 - Prob. 2E Ch. 18.2 - Prob. 3E Ch. 18.2 - Prob. 4E Ch. 18.2 - Prob. 5E Ch. 18.2 - Prob. 6E Ch. 18.2 - Prob. 7E Ch. 18.2 - Prob. 8E Ch. 18.2 - Prob. 9E Ch. 18.2 - Prob. 10E Ch. 18.2 - Prob. 11E Ch. 18.2 - Prob. 12E Ch. 18.2 - Prob. 13E Ch. 18.2 - Prob. 14E Ch. 18.2 - Prob. 15E Ch. 18.2 - Prob. 16E Ch. 18.2 - Prob. 17E Ch. 18.2 - Prob. 18E Ch. 18.2 - Prob. 19E Ch. 18.2 - Prob. 20E Ch. 18.2 - Prob. 21E Ch. 18.2 - Prob. 22E Ch. 18.2 - Prob. 23E Ch. 18.2 - Prob. 24E Ch. 18.2 - Prob. 25E Ch. 18.2 - Prob. 26E Ch. 18.2 - Prob. 27E Ch. 18.2 - Prob. 28E Ch. 18.2 - Prob. 29E Ch. 18.2 - Prob. 30E Ch. 18.2 - Prob. 31E Ch. 18.2 - Prob. 32E Ch. 18.2 - Prob. 33E Ch. 18.2 - Prob. 34E Ch. 18.2 - Prob. 35E Ch. 18.2 - Prob. 36E Ch. 18.2 - Prob. 37E Ch. 18.2 - Prob. 38E Ch. 18.3 - Prob. 1PQ Ch. 18.3 - Prob. 2PQ Ch. 18.3 - Prob. 3PQ Ch. 18.3 - Prob. 4PQ Ch. 18.3 - Prob. 5PQ Ch. 18.3 - Prob. 1E Ch. 18.3 - Prob. 2E Ch. 18.3 - Prob. 3E Ch. 18.3 - Prob. 4E Ch. 18.3 - Prob. 5E Ch. 18.3 - Prob. 6E Ch. 18.3 - Prob. 7E Ch. 18.3 - Prob. 8E Ch. 18.3 - Prob. 9E Ch. 18.3 - Prob. 10E Ch. 18.3 - Prob. 11E Ch. 18.3 - Prob. 12E Ch. 18.3 - Prob. 13E Ch. 18.3 - Prob. 14E Ch. 18.3 - Prob. 15E Ch. 18.3 - Prob. 16E Ch. 18.3 - Prob. 17E Ch. 18.3 - Prob. 18E Ch. 18.3 - Prob. 19E Ch. 18.3 - Prob. 20E Ch. 18.3 - Prob. 21E Ch. 18.3 - Prob. 22E Ch. 18.3 - Prob. 23E Ch. 18.3 - Prob. 24E Ch. 18.3 - Prob. 25E Ch. 18.3 - Prob. 26E Ch. 18.3 - Prob. 27E Ch. 18.3 - Prob. 28E Ch. 18.3 - Prob. 29E Ch. 18.3 - Prob. 30E Ch. 18.3 - Prob. 31E Ch. 18.3 - Prob. 32E Ch. 18.3 - Prob. 33E Ch. 18.3 - Prob. 34E Ch. 18.3 - Prob. 35E Ch. 18.3 - Prob. 36E Ch. 18.3 - Prob. 37E Ch. 18.3 - Prob. 38E Ch. 18.3 - Prob. 39E Ch. 18.3 - Prob. 40E Ch. 18.3 - Prob. 41E Ch. 18.3 - Prob. 42E Ch. 18.3 - Prob. 43E Ch. 18.3 - Prob. 44E Ch. 18 - Prob. 1CRE Ch. 18 - Prob. 2CRE Ch. 18 - Prob. 3CRE Ch. 18 - Prob. 4CRE Ch. 18 - Prob. 5CRE Ch. 18 - Prob. 6CRE Ch. 18 - Prob. 7CRE Ch. 18 - Prob. 8CRE Ch. 18 - Prob. 9CRE Ch. 18 - Prob. 10CRE Ch. 18 - Prob. 11CRE Ch. 18 - Prob. 12CRE Ch. 18 - Prob. 13CRE Ch. 18 - Prob. 14CRE Ch. 18 - Prob. 15CRE Ch. 18 - Prob. 16CRE Ch. 18 - Prob. 17CRE Ch. 18 - Prob. 18CRE Ch. 18 - Prob. 19CRE Ch. 18 - Prob. 20CRE Ch. 18 - Prob. 21CRE Ch. 18 - Prob. 22CRE Ch. 18 - Prob. 23CRE Ch. 18 - Prob. 24CRE Ch. 18 - Prob. 25CRE Ch. 18 - Prob. 26CRE Ch. 18 - Prob. 27CRE Ch. 18 - Prob. 28CRE Ch. 18 - Prob. 29CRE Ch. 18 - Prob. 30CRE Ch. 18 - Prob. 31CRE Ch. 18 - Prob. 32CRE Ch. 18 - Prob. 33CRE Ch. 18 - Prob. 34CRE Ch. 18 - Prob. 35CRE Ch. 18 - Prob. 36CRE Ch. 18 - Prob. 37CRE Ch. 18 - Prob. 38CRE Ch. 18 - Prob. 39CRE Ch. 18 - Prob. 40CRE Ch. 18 - Prob. 41CRE

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(a) Evaluate ∫ C F . d r considering C is the boundary of the rectangle.

Solution Summary: The author evaluates C F. d r when C is the boundary of the rectangle with vertices A, B, C and D.

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(a)

Evaluate $\int_{C} F . d r$ considering $C$ is the boundary of the rectangle.

b)

Evaluate $\int_{C} F . d r$ using stoke’s theorem when $C$ is the boundary of the wedge shaped box.

Want to see the full answer?

Chapter 18 Solutions

(a) Evaluate ∫ C F . d r considering C is the boundary of the rectangle.

Solution Summary: The author evaluates C F. d r when C is the boundary of the rectangle with vertices A, B, C and D.

Concept explainers

Videos

(a) Evaluate ∫CF.dr considering C is the boundary of the rectangle.

b) Evaluate ∫CF.dr using stoke’s theorem when C is the boundary of the wedge shaped box.

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Chapter 18 Solutions

(a)

Evaluate $\int_{C} F . d r$ considering $C$ is the boundary of the rectangle.

b)

Evaluate $\int_{C} F . d r$ using stoke’s theorem when $C$ is the boundary of the wedge shaped box.