Assignment Problems

Line Integrals

Paul Dawkins
Calculus III i

Table of Contents
Preface .................................................................................................................................................................. ii
Chapter 5 : Line Integrals ....................................................................................................................................... 3
Section 5-1 : Vector Fields .........................................................................................................................................4
Section 5-2 : Line Integrals - Part I.............................................................................................................................5
Section 5-3 : Line Integrals - Part II............................................................................................................................8
Section 5-4 : Line Integrals of Vector Fields ............................................................................................................10
Section 5-5 : Fundamental Theorem for Line Integrals ...........................................................................................13
Section 5-6 : Conservative Vector Fields .................................................................................................................15
Section 5-7 : Green's Theorem ................................................................................................................................18

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III – Assignment Problems ii

Preface
Here are a set of assignment problems for the Calculus III notes. Please note that these
problems do not have any solutions available. These are intended mostly for instructors who
might want a set of problems to assign for turning in. Having solutions available (or even just
final answers) would defeat the purpose the problems.

If you are looking for some practice problems (with solutions available) please check out the
Practice Problems. There you will find a set of problems that should give you quite a bit practice.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 3

Chapter 5 : Line Integrals

Here are a set of assignment problems for the Line Integrals chapter of the Calculus III notes. Please
note that these problems do not have any solutions available. These are intended mostly for instructors
who might want a set of problems to assign for turning in. Having solutions available (or even just final
answers) would defeat the purpose the problems.

If you are looking for some practice problems (with solutions available) please check out the Practice
Problems. There you will find a set of problems that should give you quite a bit practice.

Here is a list of all the sections for which assignment problems have been written as well as a brief
description of the material covered in the notes for that particular section.

Vector Fields – In this section we introduce the concept of a vector field and give several examples of
graphing them. We also revisit the gradient that we first saw a few chapters ago.

Line Integrals – Part I – In this section we will start off with a quick review of parameterizing curves. This
is a skill that will be required in a great many of the line integrals we evaluate and so needs to be
understood. We will then formally define the first kind of line integral we will be looking at : line
integrals with respect to arc length.

Line Integrals – Part II – In this section we will continue looking at line integrals and define the second
kind of line integral we’ll be looking at : line integrals with respect to x, y, and/or z. We also introduce
an alternate form of notation for this kind of line integral that will be useful on occasion.

Line Integrals of Vector Fields – In this section we will define the third type of line integrals we’ll be
looking at : line integrals of vector fields. We will also see that this particular kind of line integral is
related to special cases of the line integrals with respect to x, y and z.

Fundamental Theorem for Line Integrals – In this section we will give the fundamental theorem of
calculus for line integrals of vector fields. This will illustrate that certain kinds of line integrals can be
very quickly computed. We will also give quite a few definitions and facts that will be useful.

Conservative Vector Fields – In this section we will take a more detailed look at conservative vector
fields than we’ve done in previous sections. We will also discuss how to find potential functions for
conservative vector fields.

Green’s Theorem – In this section we will discuss Green’s Theorem as well as an interesting application
of Green’s Theorem that we can use to find the area of a two dimensional region.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 4

Section 5-1 : Vector Fields

  
− y2 i + x j .
1. Sketch the vector field for F =

  
2. Sketch the vector field for F = i + xy j .

  
3. Sketch the vector field for F = 4 y i + ( x + 2 ) j .

4. Compute the gradient vector field for f ( x, y ) = 6 x 2 − 9 y + x 3 y.

5. Compute the gradient vector field for f ( x, y ) = sin ( 2 x ) cos ( 3 x ) .

6. Compute the gradient vector field for f ( x, y=

, z ) z e x y + y 3 tan ( 4 x ) .

7. Compute the gradient vector field for f ( x, y, z ) = x y 2 z 3 + 4 xe y − ln ( x − z ) .

2

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 5

Section 5-2 : Line Integrals - Part I

For problems 1 – 10 evaluate the given line integral. Follow the direction of C as given in the problem
statement.

∫
1. Evaluate 3 y ds where C is the portion of x= 9 − y 2 from y = −1 and y = 2 .
C

2. Evaluate ∫ x + 2 xy ds where C is the line segment from ( 7,3) to ( 0, 6 ) .

C

∫y
2
3. Evaluate − 10 xy ds where C is the left half of the circle centered at the origin of radius 6 with
C
counter clockwise rotation.


4. Evaluate ∫x
2
− 2 y ds where C is given by r ( t ) = 4t 4 , t 4 for −1 ≤ t ≤ 0 .
C

5. Evaluate ∫z
3
− 4 x + 2 y ds where C is the line segment from ( 2, 4, −1) to (1, −1, 0 ) .
C


6. Evaluate ∫ x + 12 xz ds where C is given by r ( t ) =
C
t , 12 t 2 , 14 t 4 for −2 ≤ t ≤ 1 .

∫ z ( x + 7 ) − 2 y ds where C is the circle centered at the origin of radius 1 centered on the

3
7. Evaluate
C

x-axis at x = −3 . See the sketches below for the direction.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 6

∫
8. Evaluate 6 x ds where C is the portion of y= 3 + x 2 from x = −2 to x = 0 followed by the portion
C

of y= 3 − x 2 form x = 0 to x = 2 which in turn is followed by the line segment from ( 2, −1) to

( −1, −2 ) . See the sketch below for the direction.

9. Evaluate
C
∫ 2 − xy ds where C is the upper half of the circle centered at the origin of radius 1 with the
clockwise rotation followed by the line segment form (1, 0 ) to ( 3, 0 ) which in turn is followed by the
lower half of the circle centered at the origin of radius 3 with the clockwise rotation. See the sketch
below for the direction.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 7

10. Evaluate 3 xy + ( x − 1) ds where C is the triangle with vertices ( 0,3) , ( 6, 0 ) and ( 0, 0 ) with the
2
∫
C
clockwise rotation.

∫x
5
11. Evaluate ds for each of the following curves.
C

(a) C is the line segment from ( −1,3) to ( 0, 0 ) followed by the line segment from
( 0, 0 ) to ( 0, 4 ) .
(b) C is the portion of y= 4 − x 4 from x = −1 to x = 0 .

12. Evaluate ∫ 3x − 6 y ds for each of the following curves.

C

(a) C is the line segment from ( 6, 0 ) to ( 0,3) followed by the line segment from
( 0,3) to ( 6, 6 ) .
(b) C is the line segment from ( 6, 0 ) to ( 6, 6 ) .

∫y
2
13. Evaluate − 3 z + 2 ds for each of the following curves.
C

(a) C is the line segment from (1, 0, 4 ) to ( 2, −1,1) .

(b) C is the line segment from ( 2, −1,1) to (1, 0, 4 ) .

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 8

Section 5-3 : Line Integrals - Part II

For problems 1 – 7 evaluate the given line integral. Follow the direction of C as given in the problem
statement.

1. Evaluate ∫ xy dx + ( x − y ) dy where C is the line segment from ( 0, −3) to ( −4,1) .

C

2. Evaluate ∫ e3 x dx where C is portion of x = sin ( 4 y ) from y = π

8 to y = π .
C

3. Evaluate ∫ x dy − ( x + y ) dx where C is portion of the circle centered at the origin of radius 3 in the
2

2nd quadrant with clockwise rotation.

  2 
4. Evaluate ∫ dx − 3 y 3
dy where C is given =
by r ( t ) 4sin ( π t ) i + ( t − 1) j with 0 ≤ t ≤ 1 .
C

5. Evaluate ∫ 4y
2
dx + 3 x dy + 2 z dz where C is the line segment from ( 4, −1, 2 ) to (1, 7, −1) .
C

   
6. Evaluate ∫ ( yz + x ) dx + yz dy − ( y + z ) dz where C is given by r ( t ) =
3t i + 4sin ( t ) j + 4 cos ( t ) k
C

with 0 ≤ t ≤ π .

7. Evaluate ∫ 7 xy dy where C is the portion of=y x 2 + 5 from x = −1 to x = 2 followed by the line

C

segment from ( 2,3) to ( 4, −1) . See the sketch below for the direction.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 9

∫( y − x ) dx − 4 y dy where C is the portion of y = x from x = −2 to x = 2 followed by

2
8. Evaluate 2

the line segment from ( 2, 4 ) to ( 0, 6 ) which in turn is followed by the line segment from ( 0, 6 ) to
( −2, 4 ) . See the sketch below for the direction.

9. Evaluate ∫(x − 2 ) dx + 7 xy 2 dy for each of the following curves.

2

(a) C is the portion of x = − y 2 from y = −1 to y = 1 .

(b) C is the line segment from ( −1, −1) to (1,1) .

10. Evaluate ∫x + 9 y dy for each of the following curves.

3

(a) C is the portion of y = 1 − x 2 from x = −1 to x = 1 .

(b) C is the line segment from ( −1, 0 ) to ( 0, −1) followed by the line segment
from ( 0, −1) to (1, 0 ) .

11. Evaluate ∫ xy dx − 4 x dy for each of the following curves.

3

(a) C is the portion of the circle centered at the origin of radius 7 in the
1st quadrant with counter clockwise rotation.
(b) C is the portion of the circle centered at the origin of radius 7 in the
1st quadrant with clockwise rotation.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 10

Section 5-4 : Line Integrals of Vector Fields

     x2 y 2
2 2
(
1. Evaluate ∫ F dr where F ( x, y ) = 2 x i + y − 1 j and C is the portion of ) + = 1 that is in
25 9
C

the 1 , 4 and 3 quadrant with the clockwise orientation.

st th rd

    
2. Evaluate ∫ F  dr where F ( x , y ) = xy i + ( 4 x − 2 y ) j and C is the line segment from ( 4, −3) to
C

( 7,0 ) .
    
3. Evaluate ∫ F  dr where F ( ) ( )
( x, y ) = x3 − y i + x 2 + 7 x j and C is the portion of =
y x 3 + 2 from
C

x = −1 to x = 2 .
       
4. Evaluate ∫ F dr ( ) (
where F ( x, y )= xy i + 1 + x 2 j and C is given by r ( t )= e6t i + 4 − e 2t j for)
C

−2 ≤ t ≤ 0 .
     
5. Evaluate ∫ F  dr where F ( x , y , z ) = ( 3 x − 3 y ) i + y 3
− 10 j + y (
z k )
and C is the line segment from
C

(1, 4, −2 ) to ( 3, 4,6 ) .
     
6. Evaluate ∫ F  dr where F ( x , y , z ) = ( x + z ) i + y 3
j + (1 − x ) k and C is the portion of the spiral on
C
   
r ( t ) cos ( 2t ) i − t j + sin ( 2t ) k for −π ≤ t ≤ 2π .
the y-axis given by =

    
7. Evaluate ∫ F dr ( )
where F ( x, y ) = x 2 i + y 2 − x j and C is the line segment from ( 2, 4 ) to ( 0, 4 )
C

followed by the line segment form ( 0, 4 ) to ( 3, −1) .

     y2
8. Evaluate ∫ F dr where F ( x,=
y ) xy i − 3 j and C is the portion of x +
2
1 in the 2nd quadrant
=
C
16
with clockwise rotation followed by the line segment from ( 0, 4 ) to ( 4, −2 ) . See the sketch below.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 11

    
9. Evaluate ∫ F  dr where F ( x, y ) = xy 2 i + ( 2 y + 3x ) j and C is the portion of =
x y 2 − 1 from
C

y = −2 to y = 2 followed by the line segment from ( 3, 2 ) to ( 0,0 ) which in turn is followed by the
line segment from ( 0,0 ) to ( 3, −2 ) . See the sketch below.

    
10. Evaluate ∫ F dr (
where F ( x, y ) =− )
1 y 2 i − x j for each of the following curves.
C

(a) C is the top half of the circle centered at the origin of radius 1 with the counter
clockwise rotation.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 12

2 y2
(b) C is the bottom half of x + with clockwise rotation.
36
    
11. Evaluate ∫ F  dr where F ( x, y )= (x 2
+ y + 2 ) i + x y j for each of the following curves.
C

y x 2 − 2 from x = −3 to x = 3 .
(a) C is the portion of =
(b) C is the line segment from ( −3,5 ) to ( 3,5 ) .

    
12. Evaluate ∫ F  dr where F ( x , y )
= y 2
i + (1 − 3 x ) j for each of the following curves.
C

(a) C is the line segment from (1, 4 ) to ( −2,3) .

(b) C is the line segment from ( −2,3) to (1, 4 ) .

    
13. Evaluate ∫ F dr where F ( x, y ) =−2 x i + ( x + 2 y ) j for each of the following curves.
C

x2 y 2
(a) C is the portion of + = 1 in the 1st quadrant with counter clockwise
16 4
rotation.
x2 y 2
(b) C is the portion of + = 1 in the 1st quadrant with clockwise rotation.
16 4

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 13

Section 5-5 : Fundamental Theorem for Line Integrals

  2t
∫ by r ( t )
1. Evaluate ∇f dr where f ( x, y ) = 5 x − y 2 + 10 xy + 9 and C is given = 2
,1 − 8t with
C t +1
−2 ≤ t ≤ 0 .

 3x − 8 y    
2. Evaluate ∇f dr where f ( x, y, z ) =
∫ and C is given by r ( t ) = 6t i + 4 j + ( 9 − t 3 ) k with
C
z −6
−1 ≤ t ≤ 3 .

∫ ( x, y ) 20 y cos ( x + 3) − yx3 and C is right half of the ellipse given by
3. Evaluate ∇f dr where f =
C

( y − 1)
2

( x + 3)
2
+ 1 with clockwise rotation.
=
16
    
4. Compute ∫ F dr
C
F 2 x i + 4 y j and C is the circle centered at the origin of radius 5 with
where=
 
the counter clockwise rotation. Is ∫ dr independent of path? If it is not possible to determine if
C
F
 
∫ F dr is independent of path clearly explain why not.
C

    
5. Compute ∫
C
F  dr where F
= y i + x 2 j and C is the circle centered at the origin of radius 5 with the
   
counter clockwise rotation. Is ∫
C
F  dr independent of path? If it is not possible to determine if ∫ dr
F
C
is independent of path clearly explain why not.


6. Evaluate ∇f dr where f ( x, y, z ) = zx 2 + x ( y − 2 ) and C is the line segment from (1, 2, 0 ) to
2
∫
C

( −3,10,9 ) followed by the line segment from ( −3,10,9 ) to ( 6, 0, 2 ) .


7. Evaluate ∇f dr where f ( x, y ) =
∫ ( )
4 x + 3 xy 2 − ln x 2 + y 2 and C is the upper half of x 2 + y 2 =
1
C

( y − 2)
2

with clockwise rotation followed by the right half of ( x − 1)

2
+ 1 with counter clockwise
=
4
rotation. See the sketch below.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 14

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

 Calculus III 15

Section 5-6 : Conservative Vector Fields

For problems 1 – 4 determine if the vector field is conservative.

  
1. F = ( ) (
2 xy 3 + e x cos ( y ) i + e x sin ( y ) − 3 x 2 y 2 j )
  
2. F = ( xy 2
− 3 y 4 + 2 ) i + ( xy 2 + x 2 y 2 − x ) j

   x3 
(2 2
2 12 xy − 3x y i − 
3. F =+
2 y
− 12 x 2 y  j
 )
 

  3x 2   x3 
4. F =  8 − + 5 x 4 y 2  i +  6 + 2 − 3 y 2 + 2 x5 y  j
 y   y 

For problems 5 – 11 find the potential function for the vector field.

  2 y3    3y2  
5. F=  4 x 3 + 3 y + 3  i +  3 x − 3 y 2 − 2  j
 x   x 

  
F
6.= ( 3x e y + 4 ye x ) i − ( 7 − 2 x e y − e x ) j
2 2 4 3 2 4

  
=7. F ( cos ( x ) cos ( x + y ) − 2 y 2
− sin ( x ) sin ( x + y ) ) i − ( 4 xy + sin ( x ) sin ( x + y ) ) j

  4 2x 2    6 1 + x2  
8. F =  2 + + 2 3 i +  4 − 2  j
x y x y   xy y 

   
=9. F ( 2 xe x
2
−z
) ( 2
) (
sin ( y 2 ) − 3 y 3 i + 2 ye x − z cos ( y 2 ) − 9 xy 2 j + 12 z − e x − z sin ( y 2 ) k
2
)
   
(
10. F = 12 x − 5 z
2
) i + ln (1 + z ) j − 10 xz − 12+yzz
2
2 

k

   
( ) (
11. F = zy 2e y − x − xy 2 ze y − x i + 2 xyze y − x + xy 2 ze y − x j + xy 2e y − x − 24 z k ) ( )
    3x 2   x3 
12. Evaluate ∫ F dr where F ( x, y=
)  − 3x 2 y  i +  8 y − x3 −  j and C is the line
 y −1   ( y − 1) 
2
C 
segment from (1, 2 ) to ( 4,3) .

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 16

    
13. Evaluate ∫ F  dr where F ( x, y ) = (y 2
− 4 y + 5 ) i + ( 2 xy − 4 x − 9 ) j and C the upper half of
C
2 2
x y
+ = 1 with clockwise rotation.
36 16
    
14. Evaluate ∫ F  dr where F ( x , y ) =− 3 − (
(1 + 2 y ) e x −1
i + 3 y) (
2
+ 2e x −1
)
j and C is the portion of
C

y x3 + 1 from x = −2 to x = 1 .
=

   x    z 
15. Evaluate ∫ F  dr where F ( x,=
y, z ) i + ( 2 yz − 6 y ) j +  y 2 +  k and C is the
C x2 + z 2  x2 + z 2 
line segment from (1, 0, −1) to ( 2, −4,3) .

     
16. Evaluate ∫ F  dr where F ( x, y=) (12 xy − 2 x ) i + ( 6 x 2 − 8 yz ) j + (8 − 4 y 2 ) k and C is the spiral
C

given by r ( t ) = sin (π t ) , cos (π t ) ,3t for 0 ≤ t ≤ 6 .

    
17. Evaluate ∫ F  dr where F ( x , y ) =8 (
− 14 xy 2
+ 2 ye 2x
i + e 2x
) (
− 14 x 2
y )
j and C is the curve
C
shown below.

    
18. Evaluate ∫ F  dr where F ( x , y ) = 6(x − 5 y 2
+ 2 xy 3
− 10 i + 3 x ) (
2 2
y − 10 xy )
j and C is the curve
C
shown below.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 17

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 18

Section 5-7 : Green's Theorem

∫ ( yx − y ) dx + ( x3 + 4 ) dy where C is shown below.

2
1. Use Green’s Theorem to evaluate
C

∫ ( 7 x + y ) dy − ( x − 2 y ) dx where C is are the two circles as

2 2
2. Use Green’s Theorem to evaluate
C
shown below.

∫( y − 6 y ) dx + ( y 3 + 10 y 2 ) dy where C is shown below.

2
3. Use Green’s Theorem to evaluate
C

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 19

4. Use Green’s Theorem to evaluate ∫ xy

2
dx + (1 − xy 3 ) dy where C is shown below.
C

∫ ( y − 4 x ) dx − ( 2 + x 2 y 2 ) dy where C is shown below.

2
5. Use Green’s Theorem to evaluate
C

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 20

∫ ( y − xy 2 ) dx + ( 2 − x3 ) dy where C is shown below.

3
6. Use Green’s Theorem to evaluate
C

∫ ( 6 + x ) dx + (1 − 2 xy ) dy where C is shown below by (a) computing

2
7. Verify Green’s Theorem for
C
the line integral directly and (b) using Green’s Theorem to compute the line integral.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu

Calculus III 21

∫ ( 6 y − 3 y + x ) dx + yx3 dy where C is shown below by (a) computing

2
8. Verify Green’s Theorem for
C
the line integral directly and (b) using Green’s Theorem to compute the line integral.

© 2018 Paul Dawkins http://tutorial.math.lamar.edu