Deck 18: Line Integrals

Question 1

Let C be a segment 9 units long of the contour f(x, y)= 5.What is the work done by the gradient field of f along C?

Answer 1

0

Question 2

Given the graph of the vector field, , shown below, list the following quantities in increasing order:

Answer 2

Question 3

If the length of curve C₁ is longer than the length of curve C₂, then $\int _ { C _ { 1 } } \vec { F } \cdot \overrightarrow { d r } \geq \int _ { C _ { 2 } } \vec { F } \cdot \overrightarrow { d r }$

Answer 3

False

Question 4

Find where and C is the line from the point (2, 4, 4)to the point (0, 6, -8).

Question 5

Suppose that and .
Estimate the work done by along the line from (1, 0)to (1, 1).

Question 6

Find, by direct computation, the line integral of around the circle

Question 7

Suppose that $\int_{c} \vec{F} \cdot d \vec{r}=6$ , where C is the circle of radius 1, centered at the origin, starting at (1, 0)and traveling counter-clockwise back to (1, 0). $\int _ { G _ { 1 } } 2 \vec { F } \cdot d \vec { r } = 12$

Question 8

Calculate when and C is the line from the origin to the point (4, 4, 4).

Question 9

Explain in words and symbols how to calculate the line integral given a parameterization, of the curve C.

Question 10

Evaluate , where C is the triangular path from (0, 0)to (1, 1)to (0, 1)to (0, 0).

Question 11

Suppose that and where C₁ is the line joining (0, 0)to (1, 0), C₂ is the line joining (0, 0)to (3, 0), C₃ is the line joining (0, 0)to (0, 1)and C₄ is the line joining (0, 1)to (0, 2).
Determine, if possible, the value of the line integral of along the line from (0, 1)to (1, 0).If the value cannot be determined, say so.

Question 12

Let $\vec{F}$ be a vector field with constant magnitude $\| \vec { F } \| = 8$ Suppose that $\vec { r } ( t ) ,$ 0 $\le$ t $\le$ 5, is a parameterization of a flow line C of $\vec{F}$ .
Find $\int _ { C } \vec { F } \cdot d \vec { r }$

Question 13

Let $\vec { F } = \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { i } + \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { j }$ Is the line integral of $\vec{F}$ around the unit circle traversed counterclockwise: positive, negative, or zero?

A)Positive
B)Negative
C)Zero

Question 14

Let and let C_a be the circle of radius a centered at the origin, traveled in a counter-clockwise direction.
Find

Question 15

Let $\vec{F}$ be the constant vector field $2 \vec { i } - 2 \vec { j } + \vec { k }$ Calculate the line integral of $\vec{F}$ along a line segment L of length 9 at an angle $\pi$ /3 to $\vec{F}$

Question 16

Let and let C_a be the circle of radius a centered at the origin, traveled in a counter-clockwise direction.
Find

Question 17

Let C₁ be the rectangular loop consisting of four line segments: from (0, 0)to (1, 0), then to (1, 2), then to (0, 2), then back to (0, 0).Suppose C₂ is the triangular loop joining (0, 0)to (1, 0), then to (1, 2)then back to (0, 0), and C₃ is another triangular loop joining (0, 0)to (1, 2), then to (0, 2)and then back to (0, 0). Is it true that $\int _ { c _ { 2 } } \vec { F } \cdot \overline { d r } = \int _ { C _ { 3 } } \vec { F } \cdot \overline { d r } + \int _ { C _ { 1 } } \vec { F } \cdot \overline { d r }$ for any vector field $\vec{F}$ defined on the xy-plane?

A)Not possible to decide
B)Yes
C)No

Question 18

Consider the vector field .
Without using parametrization, calculate directly the line integral of along the line from (3, 3)to (7, 3).

Question 19

If $\int _ { C } \bar { F } \cdot \overline { d r } = 0$ , then $\vec{F}$ is perpendicular to the curve C at every point.

Question 20

Let C be the curve described by $\vec { r } ( t )$ .If the angle between $\vec { F } ( t )$ and $\vec { r } ^ { \prime } ( t )$ is less than $\pi$ /2, then $\int _ { C } \vec { F } \cdot \overline { d r } \geq 0$

Question 21

Find a vector field with the property that the line integral of along the line from (0, 0)to (a, b)is for any numbers a and b.

Question 22

Consider the vector field for certain constants A, B and C.
Use the definition of the line integral to evaluate , where C is the line from (1, 0)to (11, 1).

Question 23

Let Is path-independent?

Question 24

Given that find a function g so that Use the function g to compute where C is a curve beginning at the point (2, 4)and ending at the point (0, 1).

Question 25

Find the work done by the force field along the parabola y = 2x² from (0, 0)to (1, 2).

Question 26

Let Evaluate the line integral where C the path from (0, 0)to (1, 1)that goes along the x-axis to (1, 0), and then vertically up to (1, 1).

Question 27

If , compute where C is the curve from A(0, 0, 3)to B(2, 1, 5)shown below.
Hint: messy computation can be avoided.

Question 28

Consider the vector field for certain constants A, B and C.
Find the line integral , where C₁ is the curve

Question 29

If $\vec { F } ( x , y ) = P ( x , y ) \vec { i } + Q ( x , y ) \vec { j }$ is a gradient vector field, then $\frac { \partial P } { \partial x } = \frac { \partial Q } { \partial y }$ .

Question 30

Let Evaluate , where C is parameterized by

Question 31

Let $\vec{F}=\operatorname{grad} f$ , where $f ( x , y ) = \ln \left( 2 x ^ { 2 } + 3 y ^ { 2 } + 1 \right)$ (a)Evaluate the line integral $\int _ { C } \vec { F } \cdot \overline { d r }$ , where C is the line from (0, 0)to (2, -4).
(b)Do you expect the line integral of $\vec{F}$ along the parabola y = x²-4x, 0 $\le$ x $\le$ 2 to equal to the answer to (a)?

Question 32

Let Evaluate , where C is the line from (0, 0)to (2, 2).

Question 33

Evaluate $\int _ { - c } x \vec { i } + ( y + 10 x ) \vec { j } + ( z + 9 x ) \vec { k } \cdot \overrightarrow { d r }$ , where C is the curve $\vec { r } ( t ) = t \vec { i } + ( 1 - t ) \vec { j } + \left( t ^ { 2 } + 3 \right) \vec { k }$ for 0 $\le$ t $\le$ 1.
Note that the line integral is around -C, not C.

Question 34

Let Let , where C₁ is the line from (0, 0)to (2, 2).
Let , where C₂ is parameterized by Notice that both C₁ and C₂ go from (0, 0)to (2, 2), but is Explain.

Question 35

Let C be the curve x = 2t + cos t, y = 4t, z = 2 sin t for 0 $\le$ t $\le$ 3 $\pi$ /2.
Use a potential function to evaluate $\int_{C} \vec{F} \cdot d \vec{r}$ exactly, where $\overrightarrow { \vec { F } } = 4 x ^ { 3 } \cos ( 4 y z ) \vec { i } - 4 x ^ { 4 } z \sin ( 4 y z ) \vec { j } - 4 x ^ { 4 } y \sin ( 4 y z ) \vec { k } \text {. }$

Question 36

Let be a conservative vector field with potential function g satisfying g(0, 0)= -5.Let C₁ be the line from (0, 0)to (2, 1), C₂ the path parameterized by and C₃ the path parameterized by Suppose that and Evaluate g(2, -1).

Question 37

Consider the two-dimensional vector field Write down parameterizations of the three line segments C₁, C₂, and C₃ shown in the figure below. Use your parameterizations to compute the line integral by finding and

Question 38

Explain what is meant by saying a vector field is conservative.

A)A vector field $\vec{F}$ is called conservative if for any two points P and Q, the line integral
$\int_{C} \vec{F} \cdot d \vec{r}$ has the same value along any path C from P to Q lying in the domain of
$\vec{F}$
B)A vector field $\vec{F}$ is called conservative if for any two points P and Q, the line integral
$\int_{C} \vec{F} \cdot d \vec{r}$ has a different value along any path C from P to Q lying in the domain of
$\vec{F}$
C)A vector field $\vec{F}$ is called conservative if for any two points P and Q, the line integral
$\int_{C} \vec{F} \cdot d \vec{r}$ has the same value along a path C from P to Q lying in the domain of
$\vec{F}$
D)A vector field $\vec{F}$ is called conservative if for two specific points P and Q, the line integral
$\int_{C} \vec{F} \cdot d \vec{r}$ has the same value along any path C from P to Q lying in the domain of
$\vec{F}$

Question 39

Consider the vector field for certain constants A, B and C.
Show that is path-independent by finding its potential function.

Question 40

Let C be the circle in space with the parameterization Evaluate where

Question 41

Let and let C_a be the circle x² + y² = a² traversed counterclockwise.
Find

Question 42

The following table gives values of a function f(x, y).The table reflects the properties of the function, which is differentiable and defined for all (x, y). Let and Find if C is the circle of radius 2 centered at (4, 3).

Question 43

Let be the vector field Find using Green's theorem, if C is the unit circle traveled counterclockwise.

Question 44

Use Green's Theorem to calculate the circulation of around the triangle with vertices (0, 0), (1, 0)and (0, 1), oriented counter-clockwise.

Question 45

If C₁ and C₂ are two curves with the same starting and ending points, then $\int _ { c _ { 1 } } \vec { F } \cdot \overline { d r } = \int _ { c _ { 2 } } \vec { F } \cdot \overline { d r }$ , for any vector field $\vec{F}$

Question 46

If $\vec{F}$ is a path-independent field, then $\int _ { C } \vec { F } \cdot d \vec { r } = 0$ where C has the parameterization $\vec { r } ( t ) = \cos t \vec { i } + \sin t \vec { j } , 0 < t < 3 \pi$

Question 47

Let and be two 2-dimensional fields, where and Let C₁ be the circle with center (2, 2)and radius 1 oriented counterclockwise.
Let C₂ be the path consisting of the straight line segments from (0, 4)to (0, 1)and from (0, 1)to (3, 1).
Find the line integral Use "pi" to represent if necessary.

Question 48

Let $\vec{F}$ be the vector field shown below. Let C be the rectangular loop from (0, 0)to (1, 0)to (1, 1)to (0, 1), then back to (0, 0). Do you expect the line integral $\int_{C} \vec{F} \cdot d \vec{r}$ to be positive, negative or zero?

A)Positive
B)Negative

Question 49

Which of the two vector fields shown below is not conservative?

Question 50

Use Green's Theorem to evaluate where C is the circle of radius centered at oriented in a counter-clockwise direction.

Question 51

Let Let C₁ be the line from (0, 0)to (2, 0), C₂ the line from (2, 0)to (2,-1), C₃ the line from (2,-1)to (0,-1), and C₄ the line from (0,-1)to (0, 0).
(A)Using the definition of line integral only, without parameterizing the curves, show that the line integral of along C = C₁ + C₂ + C₃ + C₄ is -2.That is, show (B)The rectangle, R, enclosed by the lines C₁, C₂, C₃ and C₄ is of area 2.So, by Green's Theorem Is something wrong?

Question 52

Let be the vector field Find where C is the line from (0, 0)to (4, 4).

Question 53

Find the line integral of around the curve consisting of the graph of y = xⁿ from the origin to the point (1, 1), followed by straight lines from (1, 1)to (0, 1)and from (0, 1)back to the origin.

Question 54

Is the following vector field is a gradient vector field? $\vec{F}=y \vec{i}+x \vec{j}$

A)Yes
B)No

Question 55

Let C be the circular path which is the portion of the circle of radius 1 centered at the origin starting at (1, 0)and ending at (0,-1), oriented counterclockwise.
Let Determine the exact value of

Question 56

Let .
Calculate where C is the curve shown below.

Question 57

The following table gives values of a function f(x, y).The table reflects the properties of the function, which is differentiable and defined for all (x, y). Let and Find if C consists of line segments connecting (1,1), (1,10), (5,6), and (9,8)in that order.Explain your reasoning.

Question 58

Let Find a function , such that is a gradient field.

Question 59

Use Green's Theorem to find the line integral of around the closed curve composed of the graph of y = x²ⁿ where n is a positive integer and the line y = 1.

Question 60

Let Check that .Is is path-independent?

Question 61

On an exam, students were asked to evaluate , where C is the circle centered at the origin of radius r: .One student wrote:
"Since Using Green's Theorem, ."
Do you agree with the student?

Question 62

Let $\vec { F } = x ^ { 3 } \vec { i } + \left( x + \sin ^ { 3 } y \right) \vec { j }$ (a)Find the line integral $\int _ { C _ { 1 } } \vec { F } \cdot d \vec { r }$ , where C₁ is the line from (0, 0)to ( $\pi$ , 0).
(b)Evaluate the double integral $\int _ { R } 1 d A$ where R is the region enclosed by the curve y = sin x and the x-axis for 0 $\le$ x $\le$ $\pi$ .What is the geometric meaning of this integral?
(c)Use Green's Theorem and the result of part (a)to find $\int _ { c _ { 2 } } \vec { F } \cdot d \vec { r }$ where C₂ is the path from (0, 0)to ( $\pi$ , 0)along the curve y = sin x.

Question 63

Calculate the line integral of along the straight line from (3, -3)to (3, 0).

Question 64

Let $\vec{F}=\left(2+6 x e^{3\left(x^{2}+y^{2}\right)}\right) \vec{i}+\left(6 y e^{3\left(x^{2}+y^{2}\right)}\right) \vec{j}$ Use the curl test to check whether $\vec{F}$ is path-independent.

A)Path-independent
B)Not path-independent

Question 65

Let $\vec{F}=\left(3 x^{2}+y^{4} \sin x\right) \vec{i}+\left(4 y^{3} \cos x+z\right) \vec{j}-y \vec{k}$ Is the value of the line integral of $\vec{F}$ along any loop zero?

A)Yes; the function is a gradient vector field.
B)No; the function is not a gradient vector field.

Question 66

Let $\vec { F } = \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { i } + \sqrt { x ^ { 2 } + y ^ { 2 } } \vec { j }$ For a fixed $\theta$ , let C _$\theta$ be the line segment from (0, 0)to the point (cos $\theta$ , sin $\theta$ )on the unit circle.
Find a parameterization of C _$\theta$ and compute $\int_{C_{0}} \vec{F} \cdot d \vec{r}$ .(Your answer will depend on $\theta$ .)

Question 67

On an exam, students were asked to evaluate , where C has the parameterization .One student wrote:
"Using Green's Theorem, Area of the semi-circle = ."
Do you agree with the student?

Question 68

Let Find a potential function for

Question 69

Suppose a curve C is parameterized by with and suppose is a vector field for .Explain why

Question 70

Let C be the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ oriented in a counter-clockwise direction.Say whether the following statements are true or false.If $\int_{c} \vec{F} \cdot d \vec{r} \neq 0$ , we can conclude that $\vec { F }$ is not path-independent field.

Question 71

Let Find the value of where C is a path joining (0, 0)to the point (1, 2).

Question 72

State the Fundamental Theorem of Calculus for Line Integrals.

A)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.Then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
B)Suppose C is a oriented path with starting point P and end point Q.Then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
C)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.If f is a function whose gradient is continuous on the path C, then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( Q ) - f ( P )$
D)Suppose C is a piece-wise smooth oriented path with starting point P and end point Q.If f is a function whose gradient is continuous on the path C, then $\int _ { C } \operatorname { grad } f \cdot d \vec { r } = f ( P ) - f ( Q ) .$

Question 73

Let C_a be the circle x² + y² = a² oriented counter-clockwise.
Use Green's theorem to find

Question 74

Calculate the line integral of along a quarter of a circle centered at the origin, starting at (3, 0)and ending at (0, -3).

Question 75

Let f(x, y, z)be a function of three variables.Suppose that C is an oriented curve lying on the level surface f(x, y, z)= 2.
Find the line integral

Question 76

Let Show that is a gradient field by finding its potential function.

Question 77

Answer true or false, giving a reason for your answer.Let so that .
Then where C is given by the parametrization for .

Question 78

Let C be the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ oriented in a counter-clockwise direction.Say whether the following statements are true or false.
If $\int_{c} \vec{H} \cdot d \vec{r}=0$ , we can conclude that $\vec { F }$ is path-independent field.