?For the following exercises, use a computer algebra system (CAS) to evaluate the line

True or False? Line integral \(\int_C^{ }f(x,\ y)ds\) is equal to a definite integral if C is a smooth curve defined on [a, b] and if function f is continuous on some region that contains curve C. Text Transcription: \int_C^{ }f(x,\ y)ds\

True or False? Vector functions \(\mathbf{r}_{1}=t \mathbf{i}+t^{2} \mathbf{j}\), 0 ? t ? 1, and \(\mathbf{r}_{2}=(1-t) \mathbf{i}+(1-t)^{2} \mathbf{j}\), 0 ? t ? 1, define the same oriented curve. Text Transcription: \mathbf{r}_{1}=t \mathbf{i}+t^{2} \mathbf{j} \mathbf{r}_{2}=(1-t) \mathbf{i}+(1-t)^{2} \mathbf{j}

True or False? \(\int_{-C}(P d x+Q d y)=\int_{C}(P d x-Q d y)\) Text Transcription: \int_{-C}(P d x+Q d y)=\int_{C}(P d x-Q d y)

True or False? A piecewise smooth curve C consists of a finite number of smooth curves that are joined together end to end.

True or False? If C is given by x(t) = t, y(t) = t, 0 ? t ? 1, then \(\int_{C} x y d s=\int_{0}^{1} t^{2} d t\). Text Transcription: \int_{C} x y d s=\int_{0}^{1} t^{2} d t

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] \(\int_{C}(x+y) d s\) C: x = t, y = (1 ? t), z = 0 from (0, 1, 0) to (1, 0, 0) Text Transcription: \int_{C}(x+y) d s

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] \(\int_{C}(x-y) d s\) C: r(t) = 4ti + 3tj when \(0 \leq t \leq 2\) Text Transcription: int_{C}(x-y) ds 0 leq t leq 2

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] \(\int_{C}\left(x^{2}+y^{2}+z^{2}\right) d s\) C: r(t) = sin ti + cos tj + 8tk when \(0 \leq t \leq \frac{\pi}{2}\) Text Transcription: \int_{C}\left(x^{2}+y^{2}+z^{2}\right) d s 0 \leq t \leq \frac{\pi}{2}

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] Evaluate \(\int_{C} x y^{4} d s\), where C is the right half of circle \(x^{2}+y^{2}=16\) and is traversed in the clockwise direction. Text Transcription: \int_{C} x y^{4} d s x^{2}+y^{2}=16

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. [T] Evaluate \(\int_{C} 4 x^{3} d s\), where C is the line segment from (?2, ?1) to (1, 2). Text Transcription: \int_{C} 4 x^{3} d s

For the following exercises, find the work done. Find the work done by vector field F(x, y, z) = xi + 3xyj ? (x + z)k on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).

For the following exercises, find the work done. Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

For the following exercises, find the work done. Find the work done by force field \(\mathbf{F}(x,\ y,\ z)=-\frac{1}{2}x\mathbf{i}-\frac{1}{2}y\mathbf{j}+\frac{1}{4}\mathbf{k}\) on a particle as it moves along the helix r(t) = cos ti + sin tj + tk from point (1, 0, 0) to point (?1, 0, 3?). Text Transcription: \mathbf{F}(x,\ y,\ z)=-\frac{1}{2}x\mathbf{i}-\frac{1}{2}y\mathbf{j}+\frac{1}{4}\mathbf{k}

For the following exercises, find the work done. Find the work done by vector field F(x, y) = yi + 2xj in moving an object along path C, which joins points (1, 0) and (0, 1).

For the following exercises, find the work done. Find the work done by force F(x, y) = 2yi + 3xj + (x + y)k in moving an object along curve \(\mathbf{r}(t)=\cos (t) \mathbf{i}+\sin (t) \mathbf{j}+\frac{1}{6} \mathbf{k}\), where 0 ? t ? 2?. Text Transcription: \mathbf{r}(t)=\cos (t) \mathbf{i}+\sin (t) \mathbf{j}+\frac{1}{6} \mathbf{k}

For the following exercises, find the work done. Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density \(\rho(x, y)=y^{2}\). Text Transcription: \rho(x, y)=y^{2}

For the following exercises, evaluate the line integrals. Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where F(x, y) = ?1j, and C is the part of the graph of \(y=\frac{1}{2} x^{3}-x\) from (2, 2) to (?2, ?2). Text Transcription: \int_{C} \mathbf{F} \cdot d \mathbf{r} y=\frac{1}{2} x^{3}-x

For the following exercises, evaluate the line integrals. Evaluate \(\int_{\gamma}\left(x^{2}+y^{2}+z^{2}\right)^{-1} d s\), where \(\gamma\) is the helix x = cos t, y = sin t, z = t(0 ? t ? T). Text Transcription: \int_{\gamma}\left(x^{2}+y^{2}+z^{2}\right)^{-1} d s \gamma

For the following exercises, evaluate the line integrals. Evaluate \(\int_{C} y z d x+x z d y+x y d z\) over the line segment from (1, 1, 1) to (3, 2, 0). Text Transcription: \int_{C} y z d x+x z d y+x y d z

For the following exercises, evaluate the line integrals. Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral \(\int_{C} y d s \). Text Transcription: \int_{C} y d s

For the following exercises, evaluate the line integrals. [T] Use a computer algebra system to evaluate the line integral \(\int_{C} y^{2} d x+x d y\), where C is the arc of the parabola \(x=4-y^{2}\) from (?5, ?3) to (0, 2). Text Transcription: \int_{C} y^{2} d x+x d y x=4-y^{2}

For the following exercises, evaluate the line integrals. [T] Use a computer algebra system to evaluate the line integral \(\int_{C}\left(x+3 y^{2}\right) d y\) over the path C given by x = 2t, y = 10t, where 0 ? t ? 1. Text Transcription: \int_{C}\left(x+3 y^{2}\right) d y

For the following exercises, evaluate the line integrals. [T] Use a CAS to evaluate line integral \(\int_{C} x y d x+y d y\) over path C given by x = 2t, y = 10t, where 0 ? t ? 1. Text Transcription: \int_{C} x y d x+y d y

For the following exercises, evaluate the line integrals. Evaluate line integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\), where C lies along the x-axis from x = 0 to x = 5. Text Transcription: \int_{C}(2 x-y) d x+(x+3 y) d y

[T] Use a CAS to evaluate \(\int_{C} \frac{y}{2 x^{2}-y^{2}} d s\), where C is x = t, y = t, \(1 \leq t \leq 5\). Text Transcription: int_C y / 2x^2 - y^2 ds 1 leq t leq 5

[T] Use a CAS to evaluate \(\int_{C} x y d s\), where C is \(x=t^{2}\) , y = 4t, \(0 \leq t \leq 1\). Text Transcription: int_C xyds x=t^2 0 leq t leq 1

In the following exercises, find the work done by force field F on an object moving along the indicated path. F(x, y) = ?xi ? 2yj C: \(y=x^{3}\) from (0, 0) to (2, 8) Text Transcription: y=x^3

In the following exercises, find the work done by force field F on an object moving along the indicated path. F(x, y) = 2xi + yj C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)

In the following exercises, find the work done by force field F on an object moving along the indicated path. F(x, y, z) = xi + yj ? 5zk C: r(t) = 2 cos ti + 2 sin tj + tk, \(0 \leq t \leq 2 \pi\) Text Transcription: 0 leq t leq 2 pi

Let F be vector field \(\mathbf{F}(x, y)=\left(y^{2}+2 x e^{y}+1\right) \mathbf{i}+\left(2 x y+x^{2} e^{y}+2 y\right) \mathbf{j}\). Compute the work of integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where C is the path r(t) = sin ti + cos tj, \(0 \leq t \leq \frac{\pi}{2}\) . Text Transcription: F(x, y) = (y^2 + 2xe^y + 1)i + (2xy + x^2 e^y + 2y)j int_C F cdot dr 0 leq t leq pi/2

Compute the work done by force F(x, y, z) = 2xi + 3yj ? zk along path \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\), where \(0 \leq t \leq 1\). Text Transcription: r(t) = t i + t^2 j + t^3 k 0 leq t leq 1

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y)=\frac{1}{x+y} \mathbf{i}+\frac{1}{x+y} \mathbf{j}\) and C is the segment of the unit circle going counterclockwise from (1, 0) to (0, 1). Text Transcription: int_C F cdot d r F(x, y) = 1/x+y i + 1/x+y j

Force \(\mathbf{F}(x, y, z)=z y \mathbf{i}+x \mathbf{j}+z^{2} x \mathbf{k}\) acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels: a. along the path \((0,0,0) \rightarrow(1,0,0) \rightarrow(1,2,0) \rightarrow(1,2,3)\) along straight-line segments joining each pair of endpoints; b. along the straight line joining the initial and final points. c. Is the work the same along the two paths? Text Transcription: F(x, y, z) = zy i + x j + z^2 x k (0,0,0) rightarrow (1,0,0) rightarrow (1,2,0) rightarrow (1,2,3)

Find the work done by vector field F(x, y, z) = xi + 3xyj ? (x + z)k on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).

How much work is required to move an object in vector field F(x, y) = yi + 3xj along the upper part of ellipse \(\frac{x^{2}}{4}+y^{2}=1\) from (2, 0) to (?2, 0)? Text Transcription: x^2 / 4 + y^2 = 1

A vector field is given by F(x, y) = (2x + 3y)i + (3x + 2y)j. Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

Evaluate the line integral of scalar function xy along parabolic path \(y=x^{2}\) connecting the origin to point (1, 1). Text Transcription: y=x^2

Find \(\int_{C} y^{2} d x+\left(x y-x^{2}\right) d y\) along C: y = 3x from (0, 0) to (1, 3). Text Transcription: int_C y^2 dx + (xy - x^2) dy

Find \(\int_{C} y^{2} d x+\left(x y-x^{2}\right) d y\) along C: \(y^{2}=9 x\) from (0, 0) to (1, 3). Text Transcription: int_C y^2 dx + (xy - x^2) dy y^2=9x

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate \(\mathbf{F}(x, y, z)=x^{2} z \mathbf{i}+6 y \mathbf{j}+y z^{2} \mathbf{k}\), where C is represented by \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\ln t \mathbf{k}\), \(1 \leq t \leq 3\). Text Transcription: F(x, y, z) = x^2 z i + 6y j + yz^2 k r(t) = t i + t^2 j + ln t k 1 leq t leq 3

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate line integral \(\int_{\gamma} x e^{y} d s\) where, \(\gamma\) is the arc of curve \(x=e^{y}\) from (1, 0) to (e, 1). Text Transcription: int_gamma xe^y ds gamma x=e^y

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate the integral \(\int_{\gamma} x y^{2} d s\), where \(\gamma\) is a triangle with vertices (0, 1, 2), (1, 0, 3), and (0, ?1, 0). Text Transcription: int_gamma xy^2 ds gamma

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate line integral \(\int_{\gamma}\left(y^{2}-x y\right) d x\), where \(\gamma\) is curve y = ln x from (1, 0) toward (e, 1). Text Transcription: int_gamma (y^2 - xy) dx gamma

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate line integral \(\int_{\gamma} x y^{4} d s\), where \(\gamma\) is the right half of circle \(x^{2}+y^{2}=16\). Text Transcription: int_gamma xy^4 ds gamma x^2 + y^2 = 16

For the following exercises, use a CAS to evaluate the given line integrals. [T] Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y, z)=x^{2} y \mathbf{i}+(x-z) \mathbf{j}+x y z \mathbf{k}\) and C: \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+2 \mathbf{k}\), \(0 \leq t \leq 1\). Text Transcription: int_C F cdot d r F(x, y, z) = x^2 y i + (x-z) j + xyz k r(t) = t i + t^2 j + 2 k 0 leq t leq 1

For the following exercises, use a CAS to evaluate the given line integrals. Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{F}(x, y)=2 x \sin (y) \mathbf{i}+\left(x^{2} \cos (y)-3 y^{2}\right) \mathbf{j}\) and C is any path from (?1, 0) to (5, 1). Text Transcription: int_C F cdot d r F(x, y) = 2x sin (y) i + (x^2 cos (y) - 3y^2)j

Find the line integral of \(\mathbf{F}(x, y, z)=12 x^{2} \mathbf{i}-5 x y \mathbf{j}+x z \mathbf{k}\) over path C defined by \(y=x^{2}\) , \(z=x^{3}\) from point (0, 0, 0) to point (2, 4, 8). Text Transcription: F(x, y, z) = 12x^2 i - 5xy j + xz k y=x^2 z=x^3

Find the line integral of \(\int_{C}\left(1+x^{2} y\right) d s\), where C is ellipse r(t) = 2 cos ti + 3 sin tj from \(0 \leq t \leq \pi\). Text Transcription: int_C (1 + x^2 y) ds 0 leq t leq pi

For the following exercises, find the flux. Compute the flux of \(\mathbf{F}=x^{2} \mathbf{i}+y \mathbf{j}\) across a line segment from (0, 0) to (1, 2). Text Transcription: F = x^2 i + y j

For the following exercises, find the flux. Let F = 5i and let C be curve y = 0, \(0 \leq x \leq 4\). Find the flux across C. Text Transcription: 0 leq x leq 4

For the following exercises, find the flux. Let F = 5j and let C be curve y = 0, \(0 \leq x \leq 4\). Find the flux across C. Text Transcription: 0 leq x leq 4

For the following exercises, find the flux. Let F = ?yi + xj and let C: r(t) = cos ti + sin tj \((0 \leq t \leq 2 \pi)\). Calculate the flux across C. Text Transcription: (0 leq t leq 2 pi)

For the following exercises, find the flux. Let \(\mathbf{F}=\left(x^{2}+y^{3}\right) \mathbf{i}+(2 x y) \mathbf{j}\). Calculate flux F orientated counterclockwise across curve C: \(x^{2}+y^{2}=9\). Text Transcription: F = (x^2 + y^3)i + (2xy)j x^2 + y^2 = 9

Find the line integral of \(\int_{C} z^{2} d x+y d y+2 y d z\), where C consists of two parts: \(C_{1}\) and \(C_{2}\) . \(C_{1}\) is the intersection of cylinder \(x^{2}+y^{2}=16\) and plane z = 3 from (0, 4, 3) to (?4, 0, 3). \(C_{2}\) is a line segment from (?4, 0, 3) to (0, 1, 5). Text Transcription: int_C z^2 dx + ydy + 2ydz C_1 C_2 x^2 + y^2 = 16

A spring is made of a thin wire twisted into the shape of a circular helix x = 2 cos t, y = 2 sin t, z = t. Find the mass of two turns of the spring if the wire has constant mass density.

A thin wire is bent into the shape of a semicircle of radius a. If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

An object moves in force field \(\mathbf{F}(x, y, z)=y^{2} \mathbf{i}+2(x+1) y \mathbf{j}\) counterclockwise from point (2, 0) along elliptical path \(x^{2}+4 y^{2}=4\) to (?2, 0), and back to point (2, 0) along the x-axis. How much work is done by the force field on the object? Text Transcription: F(x, y, z) = y^2 i + 2(x+1) yj x^2 + 4y^2 = 4

Find the work done when an object moves in force field F(x, y, z) = 2xi ? (x + z)j + (y ? x)k along the path given by \(\mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{2}-t\right) \mathbf{j}+3 \mathbf{k}\), \(0 \leq t \leq 1\). Text Transcription: r(t) = t^2 i + (t^2 - t)j + 3k 0 leq t leq 1

If an inverse force field F is given by \(\mathbf{F}(x, y, z)=\frac{\mathbf{k}}{\|\mathbf{r}\|^{3}} \mathbf{r}\), where k is a constant, find the work done by F as its point of application moves along the x-axis from A(1, 0, 0) to B(2, 0, 0). Text Transcription: F(x, y, z) = k / |r|^3 r

David and Sandra plan to evaluate line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along a path in the xy-plane from (0, 0) to (1, 1). The force field is \(\mathbf{F}(x, y)=(x+2 y) \mathbf{i}+\left(-x+y^{2}\right) \mathbf{j}\). David chooses the path that runs along the x-axis from (0, 0) to (1, 0) and then runs along the vertical line x = 1 from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line y = x from (0, 0) to (1, 1). Whose line integral is larger and by how much? Text Transcription: int_C F cdot dr F(x, y) = (x+2y)i + (-x+y^2)j