Calculus

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 16

Integration in Vector Fields - all with Video Answers

Educators

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Section 1

Line Integrals

Problem 1

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=t \mathbf{i}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1$$

Victor Salazar

Victor Salazar

Numerade Educator

Problem 2

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad-1 \leq t \leq 1$$

Regina Hays

Numerade Educator

Problem 3

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi$$

Yiming Zhang

Numerade Educator

Problem 4

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=t \mathbf{i}, \quad-1 \leq t \leq 1$$

Regina Hays

Numerade Educator

Problem 5

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2$$

Yiming Zhang

Numerade Educator

Problem 6

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=t \mathbf{j}+(2-2 t) \mathbf{k}, \quad 0 \leq t \leq 1$$

Regina Hays

Numerade Educator

Problem 7

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad-1 \leq t \leq 1$$

Yiming Zhang

Numerade Educator

Problem 8

Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here.
$$\mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{k}, \quad 0 \leq t \leq \pi$$

Anthony Robinson

Anthony Robinson

Numerade Educator

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 $(1,0,0)$

Ekaveera Kumar

Numerade Educator

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 $(1,0,1)$

Jack Hou

Numerade Educator

Problem 11

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

Jack Hou

Numerade Educator

Problem 12

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

Jack Hou

Numerade Educator

Problem 13

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

Jack Hou

Numerade Educator

Problem 14

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

Jack Hou

Numerade Educator

Problem 15

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path from $(0,0,0)$ to $(1,1,1)$ ( see accompanying figure) given by
$$C_{1} : \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1$$
$$C_{2} : \quad \mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1$$

Victor Salazar

Numerade Educator

Problem 16

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path from $(0,0,0)$ to $(1,1,1)($ see accompanying figure) given by
$$C_{1} : \quad \mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1$$
$$C_{2} : \quad \mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$
$$C_{3} : \quad \mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$

Rebecca Polasek

Rebecca Polasek

Numerade Educator

Problem 17

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

Jack Hou

Numerade Educator

Problem 18

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

Chris Trentman

Numerade Educator

Problem 19

Evaluate $\int_{C} x d s,$ where $C$ is
a. the straight-line segment $x=t, y=t / 2,$ from $(0,0)$ to $(4,2) .$
b. the parabolic curve $x=t, y=t^{2},$ from $(0,0)$ to $(2,4)$

Carson Merrill

Numerade Educator

Problem 20

Evaluate $\int_{C} \sqrt{x+2 y} d s,$ where $C$ is
a. the straight-line segment $x=t, y=4 t,$ from $(0,0)$ to $(1,4)$ .
b. $C_{1} \cup C_{2} ; C_{1}$ is the line segment from $(0,0)$ to $(1,0)$ and $C_{2}$ is the line segment from $(1,0)$ to $(1,2)$

Jack Hou

Numerade Educator

Problem 21

Find the line integral of $f(x, y)=y e^{x^{2}}$ along the curve $\mathbf{r}(t)=4 t \mathbf{i}-3 t \mathbf{j},-1 \leq t \leq 2$

Jack Hou

Numerade Educator

Problem 22

Find the line integral of $f(x, y)=x-y+3$ along the curve $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, 0 \leq t \leq 2 \pi$

Jack Hou

Numerade Educator

Problem 23

Evaluate $\int_{C} \frac{x^{2}}{y^{4 / 3}} d s,$ where $C$ is the curve $x=t^{2}, y=t^{3},$ for $1 \leq t \leq 2$

KF

Kyle Flanagan

Numerade Educator

Problem 24

Find the line integral of $f(x, y)=\sqrt{y} / x$ along the curve $\mathbf{r}(t)=t^{3} \mathbf{i}+t^{4} \mathbf{j}, 1 / 2 \leq t \leq 1$

Jack Hou

Numerade Educator

Problem 25

Evaluate $\int_{C}(x+\sqrt{y}) d s$ where $C$ is given in the accompanying figure.

Jack Hou

Numerade Educator

Problem 26

Evaluate $\int_{C} \frac{1}{x^{2}+y^{2}+1} d s$ where $C$ is given in the accompanying figure.

Skyler Noble

Numerade Educator

Problem 27

In Exercises $27-30,$ integrate $f$ over the given curve.
$$f(x, y)=x^{3} / y, \quad C : \quad y=x^{2} / 2, \quad 0 \leq x \leq 2$$

Regina Hays

Numerade Educator

Problem 28

In Exercises $27-30,$ integrate $f$ over the given curve.
$$f(x, y)=\left(x+y^{2}\right) / \sqrt{1+x^{2}}, \quad C : \quad y=x^{2} / 2$ from (I, 1/2) to (0,0)$$

Regina Hays

Numerade Educator

Problem 29

In Exercises $27-30,$ integrate $f$ over the given curve.
$f(x, y)=x+y, \quad C : x^{2}+y^{2}=4$ in the first quadrant from (2, 0) to (0, 2)

Tanishq Gupta

Numerade Educator

Problem 30

In Exercises $27-30,$ integrate $f$ over the given curve.
$$f(x, y)=x^{2}-y, \quad C : \quad x^{2}+y^{2}=4$ in the first quadrant from $(0,2)$ to $(\sqrt{2}, \sqrt{2})$$

Tanishq Gupta

Numerade Educator

Problem 31

Find the area of one side of the "winding wall" standing orthogonally on the curve $y=x^{2}, 0 \leq x \leq 2,$ and beneath the curve on the surface $f(x, y)=x+\sqrt{y}$

HN

Harrison Nascimento

Numerade Educator

Problem 32

Find the area of one side of the "wall" standing orthogonally on the curve $2 x+3 y=6,0 \leq x \leq 6,$ and beneath the curve on the surface $f(x, y)=4+3 x+2 y$

Yiming Zhang

Numerade Educator

Problem 33

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 $\delta=(3 / 2) t$

Jack Hou

Numerade Educator

Problem 34

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.

Caleb Huber

Numerade Educator

Problem 35

Mass of with variable density 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 $(\mathbf{a}) \delta=3 t$ and $(\mathbf{b}) \delta=1$

Eric Johnson

Numerade Educator

Problem 36

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 $\delta=3 \sqrt{5}+t$

Yiming Zhang

Numerade Educator

Problem 37

Moment of inertia 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 about the $z$ -axis.

Jack Hou

Numerade Educator

Problem 38

Inertia of a 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 of the rod about the three coordinate axes.

Yiming Zhang

Numerade Educator

Problem 39

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 $I_{z}$
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 $I_{z}$ for the longer spring to be the same as that for the shorter one, or should it be different? Check your prediction by calculating $I_{z}$ for the longer spring.

Victor Salazar

Numerade Educator

Problem 40

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

Yiming Zhang

Numerade Educator

Problem 41

The arch in Example 3 Find $I_{x}$ for the arch in Example $3 .$

Tanishq Gupta

Numerade Educator

Problem 42

Center of mass and moments of inertia for wire with variable density Find the center of mass and the moments of inertia 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 $\delta=1 /(t+1)$

Yiming Zhang

Numerade Educator

Problem 43

In Exercises $43-46,$ 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 $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=\sqrt{1+30 x^{2}+10 y} ; \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+3 t^{2} \mathbf{k} 0 \leq t \leq 2$$

Tanishq Gupta

Numerade Educator

Problem 44

In Exercises $43-46,$ 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 $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=\sqrt{1+x^{3}+5 y^{3}} ; \quad \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{2} \mathbf{j}+\sqrt{t} \mathbf{k} 0 \leq t \leq 2$$

Tanishq Gupta

Numerade Educator

Problem 45

In Exercises $43-46,$ 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 $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=x \sqrt{y}-3 z^{2} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+5 t \mathbf{k} 0 \leq t \leq 2 \pi$$

Tanishq Gupta

Numerade Educator

Problem 46

In Exercises $43-46,$ 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 $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=\left(1+\frac{9}{4} z^{1 / 3}\right)^{1 / 4} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+ t^{5 / 2} \mathbf{k}, \quad 0 \leq t \leq 2 \pi$$

Tanishq Gupta

Numerade Educator