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Thomas Calculus

George B. Thomas Jr.

Chapter 16

Integrals and Vector Fields 938

Educators

YZ

Problem 1

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

Regina H.
Numerade Educator

Problem 2

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

YZ
Yiming Z.
Numerade Educator

Problem 3

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

Regina H.
Numerade Educator

Problem 4

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

YZ
Yiming Z.
Numerade Educator

Problem 5

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

Regina H.
Numerade Educator

Problem 6

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

YZ
Yiming Z.
Numerade Educator

Problem 7

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

Regina H.
Numerade Educator

Problem 8

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

YZ
Yiming Z.
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 $( 1,0,0 )$

Regina H.
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 $( 1,0,1 )$ .

YZ
Yiming Z.
Numerade Educator

Problem 11

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

Regina H.
Numerade Educator

Problem 12

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

YZ
Yiming Z.
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 $( 0 , - 1,1 )$ .

Regina H.
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 $\mathbf { r } ( t ) = t \mathbf { i } + t \mathbf { j } + t \mathbf { k } , 1 \leq t \leq \infty$

YZ
Yiming Z.
Numerade Educator

Problem 15

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

Regina H.
Numerade Educator

Problem 16

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

YZ
Yiming Z.
Numerade Educator

Problem 17

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

Regina H.
Numerade Educator

Problem 18

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

YZ
Yiming Z.
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 $( 2,4 )$

Regina H.
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 $( 1,2 )$ .

YZ
Yiming Z.
Numerade Educator

Problem 21

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

Regina H.
Numerade Educator

Problem 22

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

Bobby B.
University of North Texas

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 H.
Numerade Educator

Problem 24

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

YZ
Yiming Z.
Numerade Educator

Problem 25

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

Regina H.
Numerade Educator

Problem 26

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

YZ
Yiming Z.
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 H.
Numerade Educator

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 H.
Numerade Educator

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 H.
Numerade Educator

Problem 30

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

YZ
Yiming Z.
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 $f ( x , y ) = x + \sqrt { y }$

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

YZ
Yiming Z.
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 $\delta = ( 3 / 2 ) t$

Regina H.
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.

YZ
Yiming Z.
Numerade Educator

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

YZ
Yiming Z.
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.

Regina H.
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.

YZ
Yiming Z.
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.

Regina H.
Numerade Educator

Problem 40

Wire of constant density $\quad$ 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 $I _ { z }$

YZ
Yiming Z.
Numerade Educator

Problem 41

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

Regina H.
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 $\delta = 1 / ( t + 1 )$

YZ
Yiming Z.
Numerade Educator

Problem 43

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

Regina H.
Numerade Educator

Problem 44

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

Regina H.
Numerade Educator

Problem 45

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

Regina H.
Numerade Educator

Problem 46

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

Regina H.
Numerade Educator