Chapter 6, Line Integrals Video Solutions, Vector Calculus

Section 1

Scalar and Vector Line Integrals

Problem 1

Let $f(x, y)=x+2 y .$ Evaluate the scalar line integral $\int_{\mathbf{x}} f d s$ over the given path $\mathbf{x}$
a) $\mathbf{x}(t)=(2-3 t, 4 t-1), 0 \leq t \leq 2$
b) $\mathbf{x}(t)=(\cos t, \sin t), 0 \leq t \leq \pi$

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Problem 2

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ f(x, y, z)=x y z, \mathbf{x}(t)=(t, 2 t, 3 t), 0 \leq t \leq 2 $$

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Problem 3

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ f(x, y, z)=\frac{x+2}{y+z}, \mathbf{x}(t)=\left(t, t, t^{3 / 2}\right), 1 \leq t \leq 3 $$

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Problem 4

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ \begin{array}{l} f(x, y, z)=3 x+x y+z^{3}, \mathbf{x}(t)=(\cos 4 t, \sin 4 t, 3 t) \\ 0 \leq t \leq 2 \pi \end{array} $$

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Problem 5

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ \begin{array}{l}f(x, y, z)=\frac{z}{x^{2}+y^{2}}, \mathbf{x}(t)=\left(e^{2 t} \cos 3 t, e^{2 t} \sin 3 t, e^{2 t}\right), \\ 0 \leq t \leq 5 \end{array} $$

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Problem 6

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ \begin{array}{l} f(x, y, z)=x+y+z \\ \mathbf{x}(t)=\left\{\begin{array}{ll} (2 t, 0,0) & \text { if } 0 \leq t \leq 1 \\ (2,3 t-3,0) & \text { if } 1 \leq t \leq 2 \\ (2,3,2 t-4) & \text { if } 2 \leq t \leq 3 \end{array}\right. \end{array} $$

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Problem 7

In Exercises $2-7,$ calculate $\int_{\mathbf{x}} f d s,$ where $f$ and $\mathbf{x}$ are as indicated.
$$ \begin{array}{l} f(x, y, z)=2 x-y^{1 / 2}+2 z^{2} \\\mathbf{x}(t)=\left\{\begin{array}{ll} \left(t, t^{2}, 0\right) & \text { if } 0 \leq t \leq 1 \\ (1,1, t-1) & \text { if } 1 \leq t \leq 3 \end{array}\right. \end{array} $$

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Problem 8

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l}\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, \quad \mathbf{x}(t)=(2 t+1, t, 3 t-1), \quad 0 \leq \\ t \leq 1 \end{array} $$

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Problem 9

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l} \mathbf{F}=(y+2) \mathbf{i}+x \mathbf{j}, \quad \mathbf{x}(t)=(\sin t,-\cos t), \quad 0 \leq t \leq \\ \pi / 2 \end{array} $$

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Problem 10

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \mathbf{F}=x \mathbf{i}+y \mathbf{j}, \mathbf{x}(t)=(2 t+1, t+2), 0 \leq t \leq 1 $$

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Problem 11

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \mathbf{F}=(y-x) \mathbf{i}+x^{4} y^{3} \mathbf{j}, \mathbf{x}(t)=\left(t^{2}, t^{3}\right),-1 \leq t \leq 1 $$

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Problem 12

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l} \mathbf{F}=x \mathbf{i}+x y \mathbf{j}+x y z \mathbf{k}, \quad \mathbf{x}(t)=(3 \cos t, 2 \sin t, 5 t) \\ 0 \leq t \leq 2 \pi \end{array} $$

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Problem 13

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l} \mathbf{F}=-3 y \mathbf{i}+x \mathbf{j}+3 z^{2} \mathbf{k}, \quad \mathbf{x}(t)=\left(2 t+1, t^{2}+t, e^{t}\right), \\ 0 \leq t \leq 1 \end{array} $$

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Problem 14

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Problem 15

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l} \mathbf{F}=3 z \mathbf{i}+y^{2} \mathbf{j}+6 z \mathbf{k}, \mathbf{x}(t)=(\cos t, \sin t, t / 3), 0 \leq \\ t \leq 4 \pi \end{array} $$

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Problem 16

In Exercises $8-16,$ find $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s},$ where the vector field $\mathbf{F}$ and the path $\mathbf{x}$ are given.
$$ \begin{array}{l} \mathbf{F}=y \cos z \mathbf{i}+x \sin z \mathbf{j}+x y \sin z^{2} \mathbf{k}, \mathbf{x}(t)=\left(t, t^{2}\right. \\ \left.t^{3}\right), 0 \leq t \leq 1 \end{array} $$

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Problem 17

Determine the value of $\int_{\mathbf{x}} x d y-y d x,$ where $\mathbf{x}(t)=$ $(\cos 3 t, \sin 3 t), 0 \leq t \leq \pi$.

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Problem 18

Find the work done by the force field $\mathbf{F}=2 x \mathbf{i}+\mathbf{j}$ when a particle moves along the path $\mathbf{x}(t)=\left(t, 3 t^{2}, 2\right),$ $0 \leq t \leq 2$.

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Problem 19

If $\mathbf{x}=\left(e^{2 t} \cos 3 t, e^{2 t} \sin 3 t\right), \quad 0 \leq t \leq 2 \pi, \quad$ find
$\int_{\mathbf{x}} \frac{x d x+y d y}{\left(x^{2}+y^{2}\right)^{3 / 2}}$.

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Problem 20

If $\mathbf{x}=\left(e^{2 t} \cos 3 t, e^{2 t} \sin 3 t\right), \quad 0 \leq t \leq 2 \pi, \quad$ find
$\int_{\mathbf{x}} \frac{x d x+y d y}{\left(x^{2}+y^{2}\right)^{3 / 2}}$.

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Problem 21

Let $\mathbf{F}=\left(x^{2}+y\right) \mathbf{i}+(y-x) \mathbf{j}$ and consider the two paths
$$ \begin{array}{l} \mathbf{x}(t)=\left(t, t^{2}\right), 0 \leq t \leq 1 \text { and } \\ \mathbf{y}(t)=\left(1-2 t, 4 t^{2}-4 t+1\right), 0 \leq t \leq \frac{1}{2} \end{array} $$
(a) Calculate $\int_{\mathbf{x}} \mathbf{F} \cdot d \mathbf{s}$ and $\int_{\mathbf{y}} \mathbf{F} \cdot d \mathbf{s}$
(b) By considering the image curves of the paths $\mathbf{x}$ and $\mathbf{y},$ discuss your answers in part (a).

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Problem 22

Find the work done by the force field $\mathbf{F}=x^{2} y \mathbf{i}+z \mathbf{j}+$ $(2 x-y) \mathbf{k}$ on a particle as the particle moves along a straight line from (1,1,1) to (2,-3,3)

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Problem 23

Let $\mathbf{F}=\left(2 z^{5}-3 x y\right) \mathbf{i}-x^{2} \mathbf{j}+x^{2} z \mathbf{k} .$ Calculate the
line integral of $\mathbf{F}$ around the perimeter of the square with vertices (1,1,3),(-1,1,3),(-1,-1,3) , $(1,-1,3),$ oriented counterclockwise about the $z$ -axis.

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Problem 24

Evaluate $\int_{C}\left(x^{2}-y\right) d x+\left(x-y^{2}\right) d y,$ where $C$ is the line segment from (1,1) to (3,5)

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Problem 25

Find $\int_{C} x^{2} y d x-(x+y) d y,$ where $C$ is the trapezoid with vertices $(0,0),(3,0),(3,1),$ and $(1,1),$ oriented counterclockwise.

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Problem 26

Evaluate $\int_{C} x^{2} y d x-x y d y,$ where $C$ is the curve with equation $y^{2}=x^{3},$ from (1,-1) to (1,1)

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Problem 27

Evaluate $\int_{C} x^{2} y d x-x y d y,$ where $C$ is the curve with equation $y^{2}=x^{3},$ from (1,-1) to $(1,1) .$

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Problem 28

Evaluate $\int_{C}(x-y)^{2} d x+(x+y)^{2} d y,$ where $C$ is the portion of $y=|x|,$ from (-2,2) to (1,1) .

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Problem 29

Evaluate $\int_{C} x y^{2} d x-x y d y,$ where $C$ is the semicircular arc from (0,2) to (0,-2) traveled clockwise.

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Problem 30

Find the circulation of $\mathbf{F}=\left(x^{2}-y\right) \mathbf{i}+(x y+x) \mathbf{j}$ along the circle $x^{2}+y^{2}=16,$ oriented counterclockwise.

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Problem 31

Evaluate $\int_{C} y z d x-x z d y+x y d z,$ where $C$ is the line segment from (1,1,2) to (5,3,1) .

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Problem 32

Calculate $\int_{C} z d x+x d y+y d z,$ where $C$ is the curve obtained by intersecting the surfaces $z=x^{2}$ and $x^{2}+y^{2}=4$ and oriented counterclockwise around the $z$ -axis (as seen from the positive $z$ -axis).

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Problem 33

Show that $\int_{x} \mathbf{T} \cdot d \mathbf{s}$ equals the length of the path $\mathbf{x}$, where $T$ denotes the unit tangent vector of the path.

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Problem 34

Tom Sawyer is whitewashing a picket fence. The bases of the fenceposts are arranged in the $x y$ -plane as the quarter circle $x^{2}+y^{2}=25, x, y \geq 0,$ and the height of the fencepost at point $(x, y)$ is given by $h(x, y)=10-x-y$ (units are feet). Use a scalar line integral to find the area of one side of the fence. (See Figure $6.16 .)$

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Problem 35

Sisyphus is pushing a boulder up a 100 -ft-tall spiral staircase surrounding a cylindrical castle tower. (See Figure $6.17 .)$ (a) Suppose Sisyphus's path is described parametrically as $$ \mathbf{x}(t)=(5 \cos 3 t, 5 \sin 3 t, 10 t), \quad 0 \leq t \leq 10 $$ If he exerts a force with a constant magnitude of 50 lb tangent to his path, find the work Sisyphus does in pushing the boulder up to the top of the tower. (b) Just as Sisyphus reaches the top of the tower, he sneezes and the boulder slides all the way to the bottom. If the boulder weighs $75 \mathrm{lb}$, how much work is done by gravity when the boulder reaches the bottom?

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Problem 36

A ship is pulled through a 14 -ft-long straight channel by a line that passes from the ship around a pulley. Assume that a coordinate system is set up so that the pulley is at the point $(24,32),$ and the ship is pulled along the $y$ -axis from the origin to the point (0,14) . If the tension on the line is kept at a constant $25 \mathrm{lb}$, find the work done in tugging the ship through the channel.

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Problem 37

Suppose $C$ is the curve $y=f(x),$ oriented from $(a, f(a))$ to $(b, f(b))$ where $a<b$ and where $f$ is positive and continuous on $[a, b]$. If $\mathbf{F}=y \mathbf{i}$, show that the value of $\int_{C} \mathbf{F} \cdot d \mathbf{s}$ is the area under the graph of $f$ between $x=a$ and $x=b$.

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Problem 38

Let $\mathbf{F}$ be the radial vector field $\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$. Show that if $\mathbf{x}(t), a \leq t \leq b,$ is any path that lies on the sphere $x^{2}+y^{2}+z^{\overline{2}}=c^{2},$ then $\int_{\mathrm{x}} \mathbf{F} \cdot d \mathbf{s}=0$. (Hint: If $\mathbf{x}(t)=(x(t), y(t), z(t))$ lies on the sphere, then $[x(t)]^{2}+[y(t)]^{2}+[z(t)]^{2}=c^{2} .$ Differentiate this last equation with respect to $t .)$

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Problem 39

Let $C$ be a level set of the function $f(x, y)$. Show that $\int_{C} \nabla f \cdot d \mathbf{s}=0$

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Problem 40

You are traveling through Cleveland, famous for its lake-effect snow in winter that makes driving quite treacherous. Suppose that you are currently located 20 miles due east of Cleveland and are attempting to drive to a point 20 miles due west of Cleveland. Further suppose that if you are $s$ miles from the center of Cleveland, where the weather is the worst, you can drive at a rate of at most $v(s)=2 s+20$ miles per hour.
(a) How long will the trip take if you drive on a straight-line path directly through Cleveland? (Assume that you always drive at the maximum speed possible.)
(b) How long will the trip take if you avoid the middle of the city by driving along a semicircular path with Cleveland at the center? (Again, assume that you drive at the maximum speed possible.)
(c) Repeat parts (a) and (b), this time using $v(s)=$ $\left(s^{2} / 16\right)+25$ miles per hour as the maximum speed that you can drive.

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Problem 41

Consider a particle of mass $m$ that carries a charge $q$. Suppose that the particle is under the influence of both an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$ so that the particle's trajectory is described by the path $\mathbf{x}(t)$ for $a \leq t \leq b$. Then the total force acting on the particle is given in mks units by the Lorentz force $$ \mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}) $$ where $\mathbf{v}$ denotes the velocity of the trajectory.
(a) Use Newton's second law of motion (i.e., $\mathbf{F}=m \mathbf{a}$, where a denotes the acceleration of the particle) to show that $m \mathbf{a} \cdot \mathbf{v}=q \mathbf{E} \cdot \mathbf{v}$
(b) If the particle moves with constant speed, use part
(a) to show that $\mathbf{E}$ does no work along the path of the particle. (Hint: Apply Proposition 1.7 of Chapter 3 to $\mathbf{v}$.)

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Problem 42

Let $C$ be the segment of the parabola $y=x^{2}$ between (0,0) and (1,1) and consider the line integral $\int_{C} y^{3} d x-x^{2} d y$
(a) Let $\mathbf{x}_{0}=(0,0), \mathbf{x}_{1}=\left(\frac{1}{4}, \frac{1}{16}\right), \mathbf{x}_{2}=\left(\frac{1}{2}, \frac{1}{4}\right), \mathbf{x}_{3}=$
$\left(\frac{3}{4}, \frac{9}{16}\right), \mathbf{x}_{4}=(1,1) .$ Use the trapezoidal rule formula (7) and these points to approximate the line integral.
(b) Now calculate the exact value of the line integral and compare your results.

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Problem 43

Let $C$ be the line segment from (0,0,0) to (1,2,3) and consider the line integral $\int_{C} y z d x+(x+z) d y+$ $x^{2} y d z$
(a) Divide the segment into five regularly spaced points $\mathbf{x}_{0}, \mathbf{x}_{1}, \ldots, \mathbf{x}_{4}$ and use the trapezoidal rule formula (7) to approximate this line integral.
(b) Compare your approximation with the exact value of the line integral.

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Problem 44

Suppose that magnetic field measurements are made along a wire shaped as a curve $C$ and the results are tabulated as follows:
$$ \begin{array}{|c|c|c|} \hline k & \text { Point } \mathbf{x}_{k}= & \mathbf{B}\left(x_{k}, y_{k}, z_{k}\right)= \\
& \left(x_{k}, y_{k}, z_{k}\right) & M \mathbf{i}+N \mathbf{j}+P \mathbf{k} \\ \hline 0 & (-1,-2,-1) & \mathbf{k} \\ 1 & (0,1,-1) & \mathbf{j}+2 \mathbf{k} \\ 2 & (0,2,0) & \mathbf{i}+\mathbf{j}+2 \mathbf{k} \\ 3 & (1,2,1) & 2 \mathbf{i}+\mathbf{j}+2 \mathbf{k} \\ 4 & (1,2,2) & 2 \mathbf{i}+2 \mathbf{j}+2 \mathbf{k} \\ 5 & (1,1,2) & 2 \mathbf{i}+3 \mathbf{j}+3 \mathbf{k} \\ 6 & (1,1,1) & 3 \mathbf{i}+3 \mathbf{j}+3 \mathbf{k} \\ 7 & (1,0,0) & 4 \mathbf{i}+3 \mathbf{j}+3 \mathbf{k} \\ 8 & (0,0,0) & 4 \mathbf{i}+3 \mathbf{j}+4 \mathbf{k} \end{array} $$ By writing $\int_{C} \mathbf{B} \cdot d \mathbf{s}$ as $\int_{C} M d x+N d y+P d z,$ estimate the work done by $\mathbf{B}$ along $C$ using
(a) a trapezoidal rule approximation;
(b) a trapezoidal rule approximation using only the points $\mathbf{x}_{0}, \mathbf{x}_{2}, \mathbf{x}_{4}, \mathbf{x}_{6}, \mathbf{x}_{8}$

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