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Chapter 13

VECTOR CALCULUS

Educators


Problem 1

Evaluate the line integral, where $C$ is the given curve.
$$\int_{C} y^{3} d s, \quad C : x=t^{3}, y=t, 0 \leqslant t \leqslant 2$$

Issa D.
Issa D.
Numerade Educator

Problem 2

Evaluate the line integral, where $C$ is the given curve.
$$\int_{C} x y d s, \quad C : x=t^{2}, y=2 t, 0 \leqslant t \leqslant 1$$

Issa D.
Issa D.
Numerade Educator

Problem 3

Evaluate the line integral, where $C$ is the given curve.
$$\int_{C} x y^{4} d s, \quad C \text { is the right half of the circle } x^{2}+y^{2}=16$$

Issa D.
Issa D.
Numerade Educator

Problem 4

Evaluate the line integral, where $C$ is the given curve.
$$\int_{C} x \sin y d s, \quad C \text { is the line segment from }(0,3) \text { to }(4,6)$$

Issa D.
Issa D.
Numerade Educator

Problem 5

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{c}\left(x^{2} y^{3}-\sqrt{x}\right) d y,} \\ {C \text { is the arc of the curve } y=\sqrt{x} \text { from }(1,1) \text { to }(4,2)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 6

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{c} e^{x} d x,} \\ {C \text { is the arc of the curve } x=y^{3} \text { from }(-1,-1) \text { to }(1,1)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 7

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C}(x+2 y) d x+x^{2} d y, \quad C \text { consists of line segments }} \\ {\text { from }(0,0) \text { to }(2,1) \text { and } \text { from }(2,1) \text { to }(3,0)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 8

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x^{2} d x+y^{2} d y, \quad C \text { consists of the arc of the circle }} \\ {x^{2}+y^{2}=4 \text { from }(2,0) \text { to }(0,2) \text { followed by the line }} \\ {\text { segment from }(0,2) \text { to }(4,3)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 9

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x y z d s} \\ {C : x=2 \sin t, y=t, z=-2 \cos t, 0 \leqslant t \leqslant \pi}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 10

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x y z^{2} d s} \\ {C \text { is the line segment from }(-1,5,0) \text { to }(1,6,4)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 11

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x e^{y z} d s,} \\ {C \text { is the line segment from }(0,0,0) \text { to }(1,2,3)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 12

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C}\left(x^{2}+y^{2}+z^{2}\right) d s} \\ {C : x=t, y=\cos 2 t, z=\sin 2 t, \quad 0 \leqslant t \leqslant 2 \pi}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 13

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} x y e^{y z} d y} \\ {C : x=t, y=t^{2}, z=t^{3}, 0 \leqslant t \leqslant 1}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 14

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} y d x+z d y+x d z} \\ {C : x=\sqrt{t}, y=t, z=t^{2}, 1 \leqslant t \leqslant 4}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 15

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C} z^{2} d x+x^{2} d y+y^{2} d z} \\ {C \text { is the line segment from }(1,0,0) \text { to }(4,1,2)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 16

Evaluate the line integral, where $C$ is the given curve.
$$\begin{array}{l}{\int_{C}(y+z) d x+(x+z) d y+(x+y) d z} \\ {C \text { consists of line segments from }(0,0,0) \text { to }(1,0,1) \text { and }} \\ {\text { from }(1,0,1) \text { to }(0,1,2)}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 17

Let $\mathbf{F}$ be the vector field shown in the figure.
(a) If $C_{1}$ is the vertical line segment from $(-3,-3)$ to
$(-3,3),$ determine whether $\int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}$ is positive, nega-
tive, or zero. (b) If $C_{2}$ is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether $\int_{C_{2}} \mathbf{F} \cdot d \mathbf{r}$ is positive, negative, or zero.

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Problem 18

The figure shows a vector field $\mathbf{F}$ and two curves $C_{1}$ and $C_{2} .$ Are the line integrals of $\mathbf{F}$ over $C_{1}$ and $C_{2}$ positive, negative, or zero? Explain.

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Problem 19

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y)=x y \mathbf{i}+3 y^{2} \mathbf{j}$$
$$\mathbf{r}(t)=11 t^{4} \mathbf{i}+t^{3} \mathbf{j}, \quad 0 \leqslant t \leqslant 1$$

Issa D.
Issa D.
Numerade Educator

Problem 20

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y, z)=(x+y) \mathbf{i}+(y-z) \mathbf{j}+z^{2} \mathbf{k}$$
$$\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}+t^{2} \mathbf{k}, \quad 0 \leqslant t \leqslant 1$$

Issa D.
Issa D.
Numerade Educator

Problem 21

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y, z)=\sin x \mathbf{i}+\cos y \mathbf{j}+x z \mathbf{k}$$
$$\mathbf{r}(t)=t^{3} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k}, \quad 0 \leqslant t \leqslant 1$$

Issa D.
Issa D.
Numerade Educator

Problem 22

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+x y \mathbf{k}$$
$$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi$$

Issa D.
Issa D.
Numerade Educator

Problem 23

Use a calculator or CAS to evaluate the line integral correct to four decimal places.
$$\int_{c} \mathbf{F} \cdot d \mathbf{r},$ where $\mathbf{F}(x, y)=x y \mathbf{i}+\sin y \mathbf{j}$$ and $\mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t^{2}} \mathbf{j}, \quad 1 \leqslant t \leqslant 2$

Issa D.
Issa D.
Numerade Educator

Problem 24

Use a calculator or CAS to evaluate the line integral correct to four decimal places.
$$\begin{array}{l}{\int_{C} z e^{-x y} d s, \text { where } C \text { has parametric equations } x=t} \\ {y=t^{2}, z=e^{-t}, 0 \leqslant t \leqslant 1}\end{array}$$

Issa D.
Issa D.
Numerade Educator

Problem 25

Use a graph of the vector field $\mathbf{F}$ and the curve $C$ to guess whether the line integral of $\mathbf{F}$ over $C$ is positive, negative, or zero. Then evaluate the line integral.
$$\mathbf{F}(x, y)=(x-y) \mathbf{i}+x y \mathbf{j}$$ $C$ is the arc of the circle $x^{2}+y^{2}=4$ traversed counter clockwise from $(2,0)$ to $(0,-2)$

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Problem 26

Use a graph of the vector field $\mathbf{F}$ and the curve $C$ to guess whether the line integral of $$\mathbf{F}(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{i}+\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{j}$$ $$
C \text { is the parabola } y=1+x^{2} \text { from }(-1,2) \text { to }(1,2)$$

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Problem 27

(a) Evaluate the line integral $\int_{c} \mathbf{F} \cdot d \mathbf{r},$ where
$\mathbf{F}(x, y)=e^{x-1} \mathbf{i}+x y \mathbf{j}$ and $C$ is given by
$\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}, 0 \leqslant t \leqslant 1$
(b) Illustrate part (a) by using a graphing calculator or computer to graph $C$ and the vectors from the vector field corresponding to $t=0,1 / \sqrt{2},$ and 1 (as in Figure 13$)$ .

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Problem 28

(a) Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where
$\mathbf{F}(x, y, z)=x \mathbf{i}-z \mathbf{j}+y \mathbf{k}$ and $C$ is given by
$\mathbf{r}(t)=2 t \mathbf{i}+3 t \mathbf{j}-t^{2} \mathbf{k},-1 \leqslant t \leqslant 1 .$
(b) Illustrate part (a) by using a computer to graph $C$ and the vectors from the vector field corresponding to $t=\pm 1$ and $\pm \frac{1}{2}($ as in Figure 13$) .$

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Problem 29

Find the exact value of $\int_{C} x^{3} y^{5} d s,$ where $C$ is the part of
the astroid $x=\cos ^{3} t, y=\sin ^{3} t$ in the first quadrant.

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Problem 30

(a) Find the work done by the force field
$\mathbf{F}(x, y)=x^{2} \mathbf{i}+x y \mathbf{j}$ on a particle that moves once around the circle $x^{2}+y^{2}=4$ oriented in the counterclockwise direction.
(b) Use a computer algebra system to graph the force field and circle on the same screen. Use the graph to explain your answer to part (a).

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Problem 31

A thin wire is bent into the shape of a semicircle $x^{2}+y^{2}=4, x \geqslant 0 .$ If the linear density is a constant $k$ find the mass and center of mass of the wire.

Issa D.
Issa D.
Numerade Educator

Problem 32

A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius $a$ . If the density function is $\rho(x, y)=k x y,$ find the mass and center of mass of the wire.

Issa D.
Issa D.
Numerade Educator

Problem 33

(a) Write the formulas similar to Equations 4 for the center of mass $(\overline{x}, \overline{y}, \overline{z})$ of a thin wire in the shape of a space curve $C$ if the wire has density function $\rho(x, y, z)$ . (b) Find the center of mass of a wire in the shape of the helix $x=2 \sin t, y=2 \cos t, z=3 t, 0 \leqslant t \leqslant 2 \pi,$ if the density is a constant $k$ .

Issa D.
Issa D.
Numerade Educator

Problem 34

Find the mass and center of mass of a wire in the shape of the helix $x=t, y=\cos t, z=\sin t, 0 \leqslant t \leqslant 2 \pi,$ if the density at any point is equal to the square of the distance from the origin.

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Problem 35

If a wire with linear density $\rho(x, y)$ lies along a plane curve $C,$ its moments of inertia about the $x$ - and $y$ -axes are defined as
$$ I_{x}=\int_{C} y^{2} \rho(x, y) d s \quad I_{y}=\int_{C} x^{2} \rho(x, y) d s $$

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Problem 36

If a wire with linear density $\rho(x, y, z)$ lies along a space curve $C,$ its moments of inertia about the $x-y-$ and $z$ -axes are defined as
$$I_{x}=\int_{C}\left(y^{2}+z^{2}\right) \rho(x, y, z) d s$$
$$I_{y}=\int_{C}\left(x^{2}+z^{2}\right) \rho(x, y, z) d s$$
$$I_{z}=\int_{C}\left(x^{2}+y^{2}\right) \rho(x, y, z) d s$$
Find the moments of inertia for the wire in Exercise 33.

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Problem 37

Find the work done by the force field $\mathbf{F}(x, y)=x \mathbf{i}+(y+2) \mathbf{j}$ in moving an object along an arch of the cycloid $\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}$ 0$\leqslant t \leqslant 2 \pi$

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Problem 38

Find the work done by the force field $\mathbf{F}(x, y)=x^{2} \mathbf{i}+y e^{x} \mathbf{j}$ on a particle that moves along the parabola $x=y^{2}+1$ from $(1,0)$ to $(2,1)$

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Problem 39

Find the work done by the force field $\mathbf{F}(x, y)=x^{2} \mathbf{i}+y e^{x} \mathbf{j}$ on a particle that moves along the parabola $x=y^{2}+1$ from $(1,0)$ to $(2,1)$

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Problem 40

The force exerted by an electric charge at the origin on a charged particle at a point $(x, y, z)$ with position vector $\mathbf{r}=\langle x, y, z\rangle$ is $\mathbf{F}(\mathbf{r})=K \mathbf{r} /|\mathbf{r}|^{3}$ where $K$ is a constant.
(See Example 5 in Section $13.1 .$ ) Find the work done as the particle moves along a straight line from (2,0,0) to (2,1,5)

Issa D.
Issa D.
Numerade Educator

Problem 41

The position of an object with mass $m$ at time $t$ is
$\mathbf{r}(t)=a t t^{2} \mathbf{i}+b t^{3} \mathbf{j}, 0 \leqslant t \leqslant 1$
(a) What is the force acting on the object at time $t ?$
(b) What is the work done by the force during the time
interval 0$\leqslant t \leqslant 1 ?$

Issa D.
Issa D.
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Problem 42

An object with mass $m$ moves with position function
$$\mathbf{r}(t)=a \sin t \mathbf{i}+b \cos t \mathbf{j}+c t \mathbf{k} \quad 0 \leqslant t \leqslant \pi / 2$$
Find the work done on the object during this time period.

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Issa D.
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Problem 43

A 160 -lb man carries a $25-$ lb can of paint up a helical staircase that encircles a silo with a radius of 20 $\mathrm{ft}$ . If the silo is 90 $\mathrm{ft}$ high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?

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Issa D.
Numerade Educator

Problem 44

Suppose there is a hole in the can of paint in Exercise 43 and 9 lb of paint leaks steadily out of the can during the man's ascent. How much work is done?

Issa D.
Issa D.
Numerade Educator

Problem 45

If $C$ is a smooth curve given by a vector function $\mathbf{r}(t)$ $a \leqslant t \leqslant b,$ and $\mathbf{v}$ is a constant vector, show that
$$\int_{C} \mathbf{v} \cdot d \mathbf{r}=\mathbf{v} \cdot[\mathbf{r}(b)-\mathbf{r}(a)]$$

Issa D.
Issa D.
Numerade Educator

Problem 46

If $C$ is a smooth curve given by a vector function $\mathbf{r}(t),$ $a \leqslant t \leqslant b,$ show that $$\int_{C} \mathbf{r} \cdot d \mathbf{r}=\frac{1}{2}\left[|\mathbf{r}(b)|^{2}-|\mathbf{r}(a)|^{2}\right]
$$

Issa D.
Issa D.
Numerade Educator

Problem 47

(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle $x^{2}+y^{2}=1 .$
(b) Is this also true for a force field $\mathbf{F}(\mathbf{x})=k \mathbf{x},$ where $k$ is a constant and $\mathbf{x}=\langle x, y\rangle ?$

Issa D.
Issa D.
Numerade Educator

Problem 48

Experiments show that a steady current $I$ in a long wire produces a magnetic field $\mathbf{B}$ that is tangent to any circle that lies in the plane perpendicular to the wire and whose
center is the axis of the wire (as in the figure at the right). Ampere's Law relates the electric current to its magnetic effects and states that
$$ \int_{C} \mathbf{B} \cdot d \mathbf{r}=\mu_{0} I$$
where $I$ is the net current that passes through any surface bounded by a closed curve $C,$ and $\mu_{0}$ is a constant called the permeability of free space. By taking $C$ to be a circle
with radius $r,$ show that the magnitude $B=|\mathbf{B}|$ of the magnetic field at a distance $r$ from the center of the wire is $$B=\frac{\mu_{0} I}{2 \pi r}$$

Issa D.
Issa D.
Numerade Educator