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## 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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. Issa D.

### 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? Issa D.

### 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.

### 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.

### 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.

### 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.
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
$$\int_{C} \mathbf{B} \cdot d \mathbf{r}=\mu_{0} I$$
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 $$B=\frac{\mu_{0} I}{2 \pi r}$$ 