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Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 16

Integration In Vector Fields

Educators


Problem 1

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=t \mathbf{i}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1
$$

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Problem 2

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad-1 \leq t \leq 1
$$

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Problem 3

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi
$$

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Problem 4

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=t \mathbf{i}, \quad-1 \leq t \leq 1
$$

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Problem 5

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2
$$

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Problem 6

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=t \mathbf{j}+(2-2 t) \mathbf{k}, \quad 0 \leq t \leq 1
$$

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Problem 7

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad-1 \leq t \leq 1
$$

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Problem 8

Match the vector equations in Exercises $1-8$ with the graphs ( $a )-(h)$ given here.
Graph cannot copy
$$
\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{k}, \quad 0 \leq t \leq \pi
$$

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Problem 9

Evaluate $\int_{C}(x+y) d s$ where $C$ is the straight-line segment $x=t, y=(1-t), z=0,$ from $(0,1,0)$ to $(1,0,0)$

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Problem 10

Evaluate $\int_{C}(x-y+z-2) d s$ where $C$ is the straight-line segment $x=t, y=(1-t), z=1,$ from $(0,1,1)$ to $(1,0,1)$

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Problem 11

Evaluate $\int_{C}(x y+y+z) d s$ along the curve $\mathbf{r}(t)=2 t \mathbf{i}+$ $t \mathbf{j}+(2-2 t) \mathbf{k}, 0 \leq t \leq 1$

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Problem 12

Evaluate $\int_{C} \sqrt{x^{2}+y^{2}} d s$ along the curve $\mathbf{r}(t)=(4 \cos t) \mathbf{i}+$ $(4 \sin t) \mathbf{j}+3 t \mathbf{k},-2 \pi \leq t \leq 2 \pi$

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Problem 13

Find the line integral of $f(x, y, z)=x+y+z$ over the straight-line segment from $(1,2,3)$ to $(0,-1,1)$

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Problem 14

Find the line integral of $f(x, y, z)=\sqrt{3} /\left(x^{2}+y^{2}+z^{2}\right)$ over the curve $\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, 1 \leq t \leq \infty$.

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Problem 15

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path from $(0,0,0)$ to $(1,1,1)$ (Figure 16.6 $\mathrm{a} )$ given by
$$
\begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array}
$$
Graph cannot copy

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Problem 16

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path from $(0,0,0)$ to $(1,1,1)$ (Figure 16.6 $\mathrm{b} )$ given by
$$
\begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{3} :} & {\mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1}\end{array}
$$

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Problem 17

Integrate $f(x, y, z)=(x+y+z) /\left(x^{2}+y^{2}+z^{2}\right)$ over the path $\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, 0<a \leq t \leq b$

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Problem 18

Integrate $f(x, y, z)=-\sqrt{x^{2}+z^{2}}$ over the circle
$$
\mathbf{r}(t)=(a \cos t) \mathbf{j}+(a \sin t) \mathbf{k}, \quad 0 \leq t \leq 2 \pi
$$

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Problem 19

In Exercises $19-22,$ integrate $f$ over the given curve.
$$
f(x, y)=x^{3} / y, \quad C : \quad y=x^{2} / 2, \quad 0 \leq x \leq 2
$$

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Problem 20

In Exercises $19-22,$ integrate $f$ over the given curve.
$f(x, y)=\left(x+y^{2}\right) / \sqrt{1+x^{2}}, \quad C : \quad y=x^{2} / 2$ from $(1,1 / 2)$ to $(0,0)$

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Problem 21

In Exercises $19-22,$ integrate $f$ over the given curve.
$f(x, y)=x+y, \quad C : \quad x^{2}+y^{2}=4$ in the first quadrant from $(2,0)$ to $(0,2)$

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Problem 22

In Exercises $19-22,$ integrate $f$ over the given curve.
$f(x, y)=x^{2}-y, \quad C : \quad x^{2}+y^{2}=4$ in the first quadrant from $(0,2)$ to $(\sqrt{2}, \sqrt{2})$

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Problem 23

Mass of a wire Find the mass of a wire that lies along the curve $\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, 0 \leq t \leq 1,$ if the density is $\delta=(3 / 2) t$.

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Problem 24

Center of mass of a curved wire A wire of density $\delta(x, y, z)=15 \sqrt{y+2}$ lies along the curve $\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+$ $2 t \mathbf{k},-1 \leq t \leq 1 .$ Find its center of mass. Then sketch the curve and center of mass together.

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Problem 25

Mass of wire with variable density Find the mass of a thin wire lying along the curve $\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+\sqrt{2} t \mathbf{j}+\left(4-t^{2}\right) \mathbf{k}$ $0 \leq t \leq 1,$ if the density is (a) $\delta=3 t$ and $(\mathbf{b}) \delta=1$

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Problem 26

Center of mass of wire with variable density Find the center of mass of a thin wire lying along the curve $\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+$ $(2 / 3) t^{3 / 2} \mathbf{k}, 0 \leq t \leq 2,$ if the density is $\delta=3 \sqrt{5+t} .$

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Problem 27

Moment of inertia and radius of gyration of wire hoop $A$ circular wire hoop of constant density $\delta$ lies along the circle $x^{2}+y^{2}=a^{2}$ in the $x y$ -plane. Find the hoop's moment of inertia and radius of gyration about the $z$ -axis.

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Problem 28

Inertia and radii of gyration of slender rod A slender rod of constant density lies along the line segment $\mathbf{r}(t)=t \mathbf{j}+$ $(2-2 t) \mathbf{k}, 0 \leq t \leq 1,$ in the $y z$ -plane. Find the moments of inertia and radii of gyration of the rod about the three coordinate axes.

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Problem 29

Two springs of constant density A spring of constant density $\delta$ lies along the helix
$$
\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 \pi
$$
a. Find $I_{z}$ and $R_{z}$ .
b. Suppose that you have another spring of constant density $\delta$ that is twice as long as the spring in part (a) and lies along the helix for $0 \leq t \leq 4 \pi .$ Do you expect $I_{z}$ and $R_{z}$ for the longer spring to be the same as those for the shorter one, or should they be different? Check your predictions by calculating $I_{z}$ and $R_{z}$ for the longer spring.

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Problem 30

Wire of constant density A wire of constant density $\delta=1$ lies along the curve
$$
\mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+(2 \sqrt{2} / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq 1
$$
Find $\overline{z}, I_{z},$ and $R_{z}$

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Problem 31

The arch in Example 4 Find $I_{x}$ and $R_{x}$ for the arch in Example 4.

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Problem 32

Center of mass, moments of inertia, and radii of gyration for wire with variable density Find the center of mass, and the moments of inertia and radii of gyration about the coordinate axes of a thin wire lying along the curve
$$
\mathbf{r}(t)=t \mathbf{i}+\frac{2 \sqrt{2}}{3} t^{3 / 2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad 0 \leq t \leq 2
$$
if the density is $\delta=1 /(t+1)$

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Problem 33

In Exercises $33-36$ , use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f$ ds using Equation $(2)$ in the text.
$$
\begin{array}{l}{f(x, y, z)=\sqrt{1+30 x^{2}+10 y} ; \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+3 t^{2} \mathbf{k}} \\ {0 \leq t \leq 2}\end{array}
$$

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Problem 34

In Exercises $33-36,$ use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f$ ds using Equation $(2)$ in the text.
$$
\begin{array}{l}{f(x, y, z)=\sqrt{1+x^{3}+5 y^{3}} ; \quad \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{2} \mathbf{j}+\sqrt{t} \mathbf{k}} \\ {0 \leq t \leq 2}\end{array}
$$

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Problem 35

In Exercises $33-36,$ use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f$ ds using Equation $(2)$ in the text.
$$
\begin{array}{l}{f(x, y, z)=x \sqrt{y}-3 z^{2} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+5 t \mathbf{k}} \\ {0 \leq t \leq 2 \pi}\end{array}
$$

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Problem 36

In Exercises $33-36,$ use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f$ ds using Equation $(2)$ in the text.
$$
\begin{array}{l}{f(x, y, z)=\left(1+\frac{9}{4} z^{1 / 3}\right)^{1 / 4} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+} \\ {t^{5 / 2} \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array}
$$

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