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# Thomas Calculus

## 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 25

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

Regina H.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Problem 41

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

Regina H.

### 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.

### 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.

### 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.

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