Exam 18: Line Integrals

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?

Let $\vec{F}=81 y \vec{i}+9 x \vec{j}$ and let C_a be the circle of radius a centered at the origin, traveled in a counter-clockwise direction. Find $\lim _ { a \rightarrow 0 } \int _ { C _ { 0 } } \vec { F } \cdot d \vec { r }$

Find a vector field $\vec{F}$ with the property that the line integral of $\vec{F}$ along the line from (0, 0)to (a, b)is $3 a b ^ { 2 } + 5 a b$ for any numbers a and b.

Use Green's Theorem to find the line integral of $\vec{F}=(-x-4 y) \bar{i}+x^{2} \vec{j}$ around the closed curve composed of the graph of y = x²ⁿ where n is a positive integer and the line y = 1.

Find the line integral of $\vec{F}=(3 x-2 y) \vec{i}+(3 x+5 y) \vec{j}$ 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.

Let $\vec { F } = \left( 2 + 4 x e ^ { 2 \left( x ^ { 2 } + y ^ { 2 } \right) } \right) \vec { i } + \left( 4 y e ^ { \left. 2 x ^ { 2 } + y ^ { 2 } \right) } \right) \vec { j }$ Find the value of $\int _ { C } \vec { F } \cdot d \vec { r }$ where C is a path joining (0, 0)to the point (1, 2).

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). x y 1 2 3 4 5 6 7 8 9 10 1 7 8 4 3 8 2 1 1 5 9 2 5 9 7 11 7 3 1 4 2 10 3 6 8 14 11 10 5 13 13 12 14 4 11 15 20 22 25 24 21 21 15 12 5 17 25 30 31 32 35 37 40 35 32 6 25 30 34 30 29 26 15 14 12 9 7 39 42 51 55 50 49 47 45 35 36 8 26 21 19 24 28 27 30 33 45 39 9 49 50 55 62 69 71 60 54 49 47 10 65 70 64 6 63 49 42 41 40 38 Let $\vec{F}=\nabla f$ and $\overline{\mathrm{H}}=7 \overline{\bar{F}} .$ Find $\int_{C} \bar{H} \cdot \overline{d r}$ if C is the circle of radius 2 centered at (4, 3).

On an exam, students were asked to evaluate $\int _ { C } \left( \frac { - y \vec { i } + x \vec { j } } { x ^ { 2 } + y ^ { 2 } } \right) \cdot d \vec { r }$ , where C is the circle centered at the origin of radius r: $x = r \cos t , y = r \sin t , 0 \leq t \leq 2 \pi$ .One student wrote: "Since $\frac { \partial } { \partial y } \left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } \right) = \frac { y ^ { 2 } - x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } , \frac { \partial } { \partial x } \left( \frac { x } { x ^ { 2 } + y ^ { 2 } } \right) = \frac { y ^ { 2 } - x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } }$ Using Green's Theorem, $\int _ { C } \left( \frac { - y \vec { i } + x \vec { j } } { x ^ { 2 } + y ^ { 2 } } \right) \cdot d \vec { r } = \int _ { D } 0 d A = 0$ ." Do you agree with the student?

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 $\vec { G } = x ^ { 6 } y ^ { 3 } \vec { i } + \left( x ^ { 3 } y ^ { 6 } + 6 x \right) \vec { j }$ Determine the exact value of $\int _ { C } \vec { G } \cdot d \vec { r }$

Calculate $\int_{C} \vec{F} \cdot d \vec{r}$ when $\overrightarrow { \vec { F } } = ( y + 5 z ) \vec { i } + ( x + 4 z ) \vec { j } + 4 \vec { k }$ and C is the line from the origin to the point (4, 4, 4).

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.

Calculate the line integral of $\vec{F}=(-9 y+x) \vec{i}+2 x \vec{j}$ along a quarter of a circle centered at the origin, starting at (3, 0)and ending at (0, -3).

Let $\vec{F}=100 \vec{i}+10 \vec{j}$ and let C_a be the circle of radius a centered at the origin, traveled in a counter-clockwise direction. Find $\int _ { c _ { 0 } } \vec { F } \cdot d \vec { r }$

If $\vec { F } = 3 x ^ { 2 } \vec { i } + 4 z \sin ( 4 y z ) \vec { j } + 4 y \sin ( 4 y z ) \vec { k }$ , compute $\int_{C} \vec{F} \cdot d \vec{r}$ where C is the curve from A(0, 0, 3)to B(2, 1, 5)shown below. Hint: messy computation can be avoided.

Suppose that $\int _ { C _ { 1 } } \vec { F } \cdot \overline { d r } = 4 , \int _ { C _ { 2 } } \vec { F } \cdot \overline { d r } = 3 , \int _ { C _ { 3 } } \vec { F } \cdot \overline { d r } = 16$ and $\int _ { C _ { 4 } } \vec { F } \cdot \overline { d r } = - 1$ 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 $\vec{F}$ along the line from (0, 1)to (1, 0).If the value cannot be determined, say so.

Let C_a be the circle x² + y² = a² oriented counter-clockwise. Use Green's theorem to find $\int _ { c _ { 0 } } \left( \left( x ^ { 3 } - x ^ { 2 } y \right) \vec { i } + \left( x y ^ { 2 } + y ^ { 3 } \right) \vec { j } \right) \cdot \overrightarrow { d r }$

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.

Let $F _ { 1 } ( x , y ) = y ^ { 5 } - 10 x y + \cos y$ Find a function $F_{2}(x, y)$ , such that $\vec{F}=F_{1} \vec{i}+F_{2} \vec{j}$ is a gradient field.

Suppose that $\vec{F}(0,1)=4 \vec{i}+\vec{j}$ $\vec{F}(0.5,1)=4 \vec{j}$ $\vec { F } ( 1 , - 1 ) = 4 \vec { i } + 9 \vec { j }$ and $\vec{F}(1,0)=4 \vec{j}$ . Estimate the work done by $\vec{F}$ along the line from (1, 0)to (1, 1).

Use Green's Theorem to calculate the circulation of $\vec{F}=-5 y^{3} \vec{i}+4 x^{3} y \vec{j}$ around the triangle with vertices (0, 0), (1, 0)and (0, 1), oriented counter-clockwise.