• Home
  • Textbooks
  • Thomas Calculus
  • Integrals and Vector Fields

Thomas Calculus

George B. Thomas, Jr.

Chapter 16

Integrals and Vector Fields - all with Video Answers

Educators

+ 6 more educators

Section 1

Line Integrals

00:36

Problem 1

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

Yiming Zhang
Yiming Zhang
Numerade Educator
01:57

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
Regina Hays
Numerade Educator
07:06

Problem 3

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

Anthony Robinson
Anthony Robinson
Numerade Educator
01:57

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
Regina Hays
Numerade Educator
00:51

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
Yiming Zhang
Numerade Educator
02:05

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
Regina Hays
Numerade Educator
04:17

Problem 7

Integration in Vector Fields
$\mathbf{r}(t)=t \mathbf{j}+(2-2 t) \mathbf{k}, \quad 0 \leq t \leq 1$

Harshita Goel
Harshita Goel
Numerade Educator
07:06

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
04:26

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 $(1,0,0)$

Jack Hou
Jack Hou
Numerade Educator
01:23

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 $(1,0,1)$

Yiming Zhang
Yiming Zhang
Numerade Educator
03:22

Problem 11

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

William Semus
William Semus
Numerade Educator
03:53

Problem 12

$\begin{array}{l}{\text { Evaluate } \int_{C} \sqrt{x^{2}+y^{2}} d s \text { 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}\end{array}
$

William Semus
William Semus
Numerade Educator
03:25

Problem 13

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

William Semus
William Semus
Numerade Educator
04:21

Problem 14

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

Melissa Munoz
Melissa Munoz
Numerade Educator
10:19

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
$$
\begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1}\end{array}
$$
The paths of integration for Exercises 15 and 16

Lucas Finney
Lucas Finney
Numerade Educator
View

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
$$
\begin{array}{ll}{C_{1} :} & {\mathbf{r}(t)=t \mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{2} :} & {\mathbf{r}(t)=t \mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1} \\ {C_{3} :} & {\mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1}\end{array}
$$
The paths of integration for Exercises 15 and 16

Victor Salazar
Victor Salazar
Numerade Educator
02:46

Problem 17

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

Melissa Munoz
Melissa Munoz
Numerade Educator
05:50

Problem 18

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

Harshita Goel
Harshita Goel
Numerade Educator
03:30

Problem 19

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

Diogo Caetano
Diogo Caetano
Numerade Educator
View

Problem 20

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

Victor Salazar
Victor Salazar
Numerade Educator
01:19

Problem 21

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

Yiming Zhang
Yiming Zhang
Numerade Educator
02:08

Problem 22

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

Diogo Caetano
Diogo Caetano
Numerade Educator
04:39

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
$1 \leq t \leq 2$

Regina Hays
Regina Hays
Numerade Educator
01:19

Problem 24

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

Yiming Zhang
Yiming Zhang
Numerade Educator
02:57

Problem 25

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

Linda Hand
Linda Hand
Numerade Educator
02:38

Problem 26

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

Lucas Finney
Lucas Finney
Numerade Educator
04:11

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
Regina Hays
Numerade Educator
03:52

Problem 28

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

Regina Hays
Regina Hays
Numerade Educator
03:42

Problem 29

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

Regina Hays
Regina Hays
Numerade Educator
02:38

Problem 30

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

Narayan Hari
Narayan Hari
Numerade Educator
05:32

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 $f(x, y)=x+\sqrt{y}$

Regina Hays
Regina Hays
Numerade Educator
04:07

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 $f(x, y)=4+3 x+2 y .$

Jack Hou
Jack Hou
Numerade Educator
05:37

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 $\delta=(3 / 2) t$

Dale Sanford
Dale Sanford
Numerade Educator
10:05

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.

Jack Hou
Jack Hou
Numerade Educator
07:22

Problem 35

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 (b) $\delta=1 .$

Regina Hays
Regina Hays
Numerade Educator
02:08

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
Yiming Zhang
Numerade Educator
04:31

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.

Veronica Santana
Veronica Santana
Numerade Educator
04:31

Problem 38

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.

Veronica Santana
Veronica Santana
Numerade Educator
View

Problem 39

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

Victor Salazar
Victor Salazar
Numerade Educator
01:37

Problem 40

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

Yiming Zhang
Yiming Zhang
Numerade Educator
03:22

Problem 41

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

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
02:49

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
Yiming Zhang
Numerade Educator
View

Problem 43

In Exercises $43-46,$ use a CAS to perform the following steps to
evaluate the line integrals.
$$
\begin{array}{l}{\text { a. Find } d s=|\mathbf{v}(t)| d t \text { for the path } \mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k} \text { . }} \\ {\text { b. Express the integrand } f(g(t), h(t), k(t))|\mathbf{v}(t)| \text { as a function of }} \\ {\text { the parameter } t .} \\ {\text { c. Evaluate } \int_{C} f d s \text { using Equation }(2) \text { in the text. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y, z)=\sqrt{1+30 x^{2}+10 y} ; \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+3 t^{2} \mathbf{k}} \\ {0 \leq t \leq 2}\end{array}
$$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
View

Problem 44

In Exercises $43-46,$ use a CAS to perform the following steps to
evaluate the line integrals.
$$
\begin{array}{l}{\text { a. Find } d s=|\mathbf{v}(t)| d t \text { for the path } \mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k} \text { . }} \\ {\text { b. Express the integrand } f(g(t), h(t), k(t))|\mathbf{v}(t)| \text { as a function of }} \\ {\text { the parameter } t .} \\ {\text { c. Evaluate } \int_{C} f d s \text { using Equation }(2) \text { in the text. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y, z)=\sqrt{1+x^{3}+5 y^{3}} ; \quad \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{2} \mathbf{j}+\sqrt{t} \mathbf{k}} \\ {0 \leq t \leq 2}\end{array}
$$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:18

Problem 45

In Exercises $43-46,$ use a CAS to perform the following steps to
evaluate the line integrals.
$$
\begin{array}{l}{\text { a. Find } d s=|\mathbf{v}(t)| d t \text { for the path } \mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k} \text { . }} \\ {\text { b. Express the integrand } f(g(t), h(t), k(t))|\mathbf{v}(t)| \text { as a function of }} \\ {\text { the parameter } t .} \\ {\text { c. Evaluate } \int_{C} f d s \text { using Equation }(2) \text { in the text. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y, z)=x \sqrt{y}-3 z^{2} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+5 t \mathbf{k}} \\ {0 \leq t \leq 2 \pi}\end{array}
$$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:18

Problem 46

In Exercises $43-46,$ use a CAS to perform the following steps to
evaluate the line integrals.
$$
\begin{array}{l}{\text { a. Find } d s=|\mathbf{v}(t)| d t \text { for the path } \mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k} \text { . }} \\ {\text { b. Express the integrand } f(g(t), h(t), k(t))|\mathbf{v}(t)| \text { as a function of }} \\ {\text { the parameter } t .} \\ {\text { c. Evaluate } \int_{C} f d s \text { using Equation }(2) \text { in the text. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y, z)=\left(1+\frac{9}{4} z^{1 / 3}\right)^{1 / 4} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+} \\ {t^{5 / 2} \mathbf{k}, \quad 0 \leq t \leq 2 \pi}\end{array}
$$

Tanishq Gupta
Tanishq Gupta
Numerade Educator