Created by T. Madas LINE INTEGRALS. Created by T. Madas

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1 LINE INTEGRALS

2 LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES

3 Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( ) 6,

4 Question 2 It is given that the vector function F satisfies ( x 2 y 2 ) ( 2xy) F = i + j. Evaluate the line integral ( 4,2) F i dr, ( 2, 1) along a path joining directly the points with Cartesian coordinates ( 2,1) and ( 4,2 ). 30

5 Question 3 The path along the straight line with equation y x 2 denoted by C. a) Evaluate the integral 3 3 ( x + y) dx + ( x y ) dy. C = +, from A ( 0,2) to B ( 3,5) b) Show that the integral is independent of the path chosen from A to B. c) Verify the independence of the path by evaluating the integral of part (a) along a different path from A to B., is 117

6 Question 4 The path along the perimeter of the triangle with vertices at ( 0,0 ), ( 1,0 ) and ( 0,1 ), is denoted by C. Evaluate the integral 2 x dx 2xy dy. C 1 3

7 Question 5 The path along the perimeter of the triangle with vertices at ( 0,0 ), ( 1,0 ) and ( 0,1 ), is denoted by C. Evaluate the integral 2 2 ( x + x) dx + ( x y) dy. C 5 12

8 Question 6 The functions F and G are defined as 2 (, ) = x y and G( x, y) = ( x + y) 2 F x y The anticlockwise path along the perimeter of the triangle whose vertices are located 0,1, is denoted by C. at ( 0,0 ), ( 1,0 ) and ( ) Evaluate the line integral F dx + G dy. C 7 12

9 Question 7 The anticlockwise path along the perimeter of the square whose vertices are located at 0,1, is denoted by C. the points ( 0,0 ), ( 1,0 ), ( 1,1 ) and ( ) Evaluate the line integral 2 ( x + xy) dx + ( x + y) 3 dy. You may not use Green s theorem in this question. C 3

10 Question 8 Evaluate the integral ( 5,0) ( 3y) dx + ( 3x + 2y) dy, ( 1,7 ) along a path joining the points with Cartesian coordinates ( 1,7 ) and ( 5,0 ). 28

11 Question 9 Evaluate the integral ( 3,4) ( 3 x y ) dx + ( 2 x y ) dy, ( 1,1 ) along a path joining the points with Cartesian coordinates ( 1,1 ) and ( 3,4 ). 431

12 Question 10 ( x, y) ( 2xy 2 + cos x) + ( 2x 2 y sin y) F i j. Show that the vector field F is conservative, and hence evaluate the integral C Fi dr, where C is the arc of the circle with equation x π + y =, y 0, 4 π from A,0 2 to B π,0 2. 2

13 Question 11 In this question α, β and γ are positive constants. 2 2 x β x y F = α y e + e. A particle of mass m is moving on the x-y plane, under the action of F. Find the work done by F on the particle in moving it from the Cartesian origin O to the point ( 1,1 ), in each of the following cases. a) Directly from O to ( 1,0 ), then directly from ( 1,0 ) to ( 1,1 ). b) Directly from O to ( 0,1 ), then directly from ( 0,1 ) to ( 1,1 ). c) Moving the particle with velocity = γ ( x + y ) v e e. W1 = β, W2 = α, W3 = 1 3 ( β α )

14 Question 12 Evaluate the line integral y x dx x dy, C ( 1) e x x + + ( e + 1) where C is a circle of radius 1, centre at the origin O, traced anticlockwise. π

15 Question 13 It is given that the vector function F satisfies 3 3 ( sin x xy) ( x y sin y) F = i + + j. Evaluate the line integral C Fidr, where C is the ellipse with Cartesian equation 2 2 2x + 3y = 2y. π 3 6

16 Question 14 It is given that the vector function F satisfies Evaluate the line integral 3 [ xcos x] 15xy ln ( 1 y ) F = i j. where C is the curve C Fidr, {( ) } ( ) 2 { } x, y : y = 3, 2 x 2 x, y : y = x 1, 2 x 2, traced in an anticlockwise direction. 224

17 LINE INTEGRALS 2 DIMENSIONAL PARAMETERIZATIONS

18 Question 1 The path along the semicircle with equation x y = 1, x 0 from A ( 0,1) to ( 0, 1) B, is denoted by C. Evaluate the integral 3 3 ( x + y ) dx. C 3 8 π

19 Question 2 Evaluate the integral ( 6,12 ) 2 ( 6x 2xy) ds, ( 0,0) where s is the arclength along the straight line segment from ( 0,0 ) to ( 6,12 ). MM2-B, 144 5

20 Question 3 Evaluate the integral ( 3,3) ( y + x) dx + ( y x) dy, ( 1,0) along the curve with parametric equations 2 x = 2t 3t + 1 and 2 y = t

21 Question 4 Evaluate the line integral ( 0,5) ( 2x + y) ds, ( 5,0) where s is the arclength along the quarter circle with equation x y =

22 Question 5 Evaluate the line integral 5 y dx, C where C is a circle of radius 2, centre at the origin O, traced anticlockwise. You may not use Green s theorem in this question. 40π

23 Question 6 Evaluate the line integral 3 y dx + ( xy) dy, C where C is a circle of radius 1, centre at the origin O, traced anticlockwise. You may not use Green s theorem in this question. MM2-D, 3π 4

24 Question 7 Evaluate the line integral y dx + x( 2 + y) dy, C where C is a circle of radius 1, centre at the origin O, traced anticlockwise. You may not use Green s theorem in this question. π

25 Question 8 Evaluate the line integral 2 x y dx, 2 2 x + y C where C is a circle of radius 1, centre at the origin O, traced anticlockwise. π 4

26 Question 9 A y 2 2 ( x 1) = 16( 1 y ) O B x The figure above shows the ellipse with equation 2 2 ( x 1) 16( 1 y ) =. The ellipse meets the positive x and y axes at the points A and B, respectively, as shown in the figure. The elliptic path C is the clockwise section from A to B. Determine the value of each of the following line integrals. x xy dx y 1 x dy. 2 a) ( + ) + ( + ) C y dx + x 1 dy. 3 b) ( ) 3 C MM2C, , [solution overleaf]

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28 Question 10 The closed curve C bounds the finite region R in the x-y plane defined as 2 2 (, ) { 0 0 1} R x y = x + y x y x + y. Evaluate the line integral where C is traced anticlockwise. 2 ( xy dx + x dy ), C 0

29 Question 11 Evaluate the line integral 2 2 ( x ) + ln ( + ) y arctan dx x y dy, C where C is the polar rectangle such that 1 r 2, 0 θ π, traced anticlockwise. π

30 Question 12 Evaluate the line integral ( 2x y) dx + ( 2y x) dy, where C is an ellipse with Cartesian equation C 2 2 x y + = 1, 9 4 traced anticlockwise. You may not use Green s theorem in this question. 0

31 Question 13 It is given that the vector function F satisfies ( x 3y) ( y 2x) F = i + j. Evaluate the line integral C Fidr, where C is the ellipse with cartesian equation 2 2 x y + = You may not use Green s theorem in this question. 6π

32 LINE INTEGRALS IN 3 DIMENSIONS

33 Question 1 It is given that the vector function F satisfies ( x 2 y) ( 4 xy 2 ) ( 6 xz) F = i + j + k. Evaluate the line integral ( 10,4,8) F i dr, dr = ( dx, dy, dz )T, ( 0,0,0) along a path given by the parametric equations x = 5t, 2 y = t, 3 z = t

34 Question 2 It is given that the vector function F satisfies ( x 2 y ) ( xy 2 ) ( yz ) F = i + j+ k. Evaluate the line integral ( 1,2,3) F i dr, dr = ( dx, dy, dz )T, ( 0,0,0) along a path of three straight line segments joining ( 0,0,0) to ( 1,0,0 ), ( ) ( 1,2,0 ) and ( 1,2,0 ) to ( 1,2,3 ). 1,0,0 to 32 3

35 Question 3 It is given that (,, ) F x y z j r, where r = x i + y j + zk. Evaluate the line integral C Fi dr, where C is the closed curve given parametrically by ( t) = ( t t 2 ) + ( 2t 2t 2 ) + ( t 2 t 3 ) R i j k, 0 t

36 Question 4 The simple closed curve C has Cartesian equation x y = 4, z = 3. Given that F = x z i + y x j+ z yk, evaluate the integral C F i dr. You may not use Green s theorem in this question. 4π

37 Question ( xz y) ( xy z) ( x y z ) F = i + + j+ + + k. Determine the work done by F, when it moves in a complete revolution in a circular path of radius 2 around the z axis, at the level of the plane with equation z = 6. You may not use Green s theorem in this question. 4π

38 Question 6 Evaluate the integral ( 5,3,4) 2 ( 3 x 2 y ) dx + ( y + z ) dy + ( 1 z ) dz, ( 1,1,0 ) along the straight line segment joining the points with Cartesian coordinates ( 1,1,0 ) and ( 5,3,4 ). MM2A, 32 3

39 Question 7 ( ) 2 2 F x, y, z yz i + xz j+ 2xyzk. Show that the vector field F is conservative, and hence evaluate the integral ( 3,5,10 ) F i dr. ( 1,1,4 ) 1484

40 Question 8 A vector field F is defined as 2 ( x, y, z) [ x + yz] + [ y + xz] + x( y + 1) + z F i j k. The closed path C joins ( 0,0,0 ) to ( 1,1,1 ), ( 1,1,1 ) to ( 1,1,0 ), ( 1,1,0 ) to ( 0,0,0 ), in that order. By writing ( x, y, z) = ( x, y, z) + ( x, y, z) F G H, = G for some smooth for some vector functions G and H, where g ( x, y, z) ( x, y, z) scalar function g ( x, y, z ), evaluate the line integral C Fidr. 1 2

41 Question 9 A vector field F is defined as ( x, y, z) ( yz + y 2 ) + ( xz + 2xy) + ( xy + 4z 3 ) F i j k. a) Show that F is conservative. b) Hence evaluate the integral ( 1,1,1 ) F i dr. ( 0,0,0) 3

42 Question 10 A curve C is defined as ( x, y, z) ( cos3 t,sin 3 t, t) =, 0 t 2π. a) Sketch the graph of C. (,, ) F x y z xy i + yz j+ zxk. b) Determine whether the vector field F is conservative. c) Evaluate the integral C Fi dr. π 2

43 Question 11 Evaluate the integral ( 2,0,1) ( 3x yz + 6x) dx + ( x z 8y) dy + ( x y + 1) dz, ( 1,2,3) along a path joining the points with Cartesian coordinates ( 1,2,3 ) and ( 2,0,1 ). 29

44 Question 12 A curve C is defined by = ( t) r r, 0 t 2π as Evaluate the integral ( ) ( ) ( ) r t = x, y, z = 2 t sin t, 3 cos t, 1+ cost. z ds, C where s is the arclength along C. 32 3

45 Question 13 A vector field F and a scalar field ψ are given. Evaluate the integral ( 3 3 x y 13 ) ( 15 z ) ( xz) F = i + j k and ψ 96 ( x, y, z) = xe z. ( 2,4,64) F ( 0,0,0) [ + ψ ] i dr, 2 y along the curve with parametric equations x = t, y = t and 3 z = t. 2e 288

46 Question 14 It is given that the vector function F satisfies ( 3 x 2 yz 2 z) ( x 3 z 2 y) ( x 3 y 2 x) F = + i + + j+ + k. Evaluate the line integral ( 4,0,1) ( F dr 2,2,0) i, along a path joining the points with Cartesian coordinates ( 1,2,3 ) and ( 2,0,1 ). 4

47 Question 15 It is given that the vector function F satisfies ( 1 xy 2 ) ( x xyz) ( y sin z) F = + i + + j+ k. Evaluate the line integral C Fidr, where C is the anticlockwise cartesian path x y = 16, z = 3. You may not use Green s theorem in this question. 16π

48 Question 16 Evaluate the line integral 2 x dx + ( x 2yz) dy + ( x + z) dz, C where C is the intersection of the surfaces with respective Cartesian equations x + y + z = 1, z 0 and 2 2 x + y = x, z 0. π 4

49 Question 17 It is given that the vector field F satisfies F = 8z i + 4x j+ yk. Evaluate the line integral C Fidr, where C is the intersection of the surfaces with respective Cartesian equations 2 2 z = x + y and z = y. You may not use Stokes Theorem in this question. π

50 Question 18 It is given that the vector field F satisfies F = y i + z j+ x k. Evaluate the line integral C Fidr, where C is the intersection of the surfaces with respective Cartesian equations x + y + z = 1, z 0 and 2 2 x + y = x, z 0. You may not use Stokes Theorem in this question. π 4 [solution overleaf]

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52 LINE INTEGRALS IN POLAR COORDINATES

53 Question 1 O initial line The figure above shows the closed curve C with polar equation π r = sin 2θ, 0 θ. 2 The vector field F is given in plane polar coordinates ( r, θ ) by ( r, θ ) = ( r 2 cosθ sinθ ) ˆ + ( r cosθ ) F r ˆθ. Evaluate the line integral C F i dr. 8 15

54 Question 2 O initial line The figure above shows the curve C with polar equation r = θ, 0 θ 2π. The vector field F is given in Cartesian coordinates by (, ) F x y = xi + y j. Evaluate the line integral C F i dr. 2 2π

55 Question 3 O initial line The figure above shows the closed curve C with polar equation r = 1+ cosθ, 0 θ 2π. The vector field F is given in Cartesian coordinates by (, ) F x y = y i + x j. Evaluate the line integral C F i dr. 3π

56 Question 48 O initial line The figure above shows the closed curve C with polar equation r = 3 + sinθ, 0 θ 2π. The vector field F is given in Cartesian coordinates by ( x, y) = ( x + y) + ( x + y) F i j. Evaluate the line integral C F i dr. MM2-E, 19π