University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 15

Integrals and Vector Fields

Educators


Problem 1

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

Jack H.
Numerade Educator

Problem 2

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

Jack H.
Numerade Educator

Problem 3

Match the vector equations with the graphs (a)-(h) given here. (GRAPH CANT COPY)
$$\mathbf{r}(t)=(2 \cos t) \mathbf{1}+(2 \sin t) \mathbf{J}, \quad 0 \leq t \leq 2 \pi$$

Jack H.
Numerade Educator

Problem 4

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

Jack H.
Numerade Educator

Problem 5

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

Jack H.
Numerade Educator

Problem 6

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

Jack H.
Numerade Educator

Problem 7

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

Jack H.
Numerade Educator

Problem 8

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

Jack H.
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).

Jack 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).

Jack H.
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 H.
Numerade Educator

Problem 12

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

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

Jack H.
Numerade Educator

Problem 15

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path $C_{1}$ followed by $C_{2}$ from (0,0,0) to (1,1,1) (see accompanying figure) given by
$$\begin{array}{ll}C_{1}: & \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}, \quad 0 \leq t \leq 1 \\C_{2}: & \mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 1\end{array}$$
(GRAPH CANT COPY)
The paths of integration for Exercises 15 and 16

Jack H.
Numerade Educator

Problem 16

Integrate $f(x, y, z)=x+\sqrt{y}-z^{2}$ over the path $C_{1}$ followed by $C_{2}$ followed by $C_{3}$ 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$$$C_{3}: \quad \mathbf{r}(t)=t \mathbf{i}+\mathbf{j}+\mathbf{k}, \quad 0 \leq t \leq 1$$
(GRAPH CANT COPY)

Jack H.
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 H.
Numerade Educator

Problem 18

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

Jack H.
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).

Jack 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).

Jack H.
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 H.
Numerade Educator

Problem 22

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

Jack H.
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 $1 \leq t \leq 2$.

Jack H.
Numerade Educator

Problem 24

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

Jack H.
Numerade Educator

Problem 25

Evaluate $\int_{C}(x+\sqrt{y}) d s,$ where $C$ is given in the accompanying figure. (FIGURE CAN'T COPY)

Jack 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. (FIGURE CAN'T COPY)

Jack H.
Numerade Educator

Problem 27

Integrate $f$ over the given curve.
$f(x, y)=x^{3} / y, \quad C: \quad y=x^{2} / 2, \quad 0 \leq x \leq 2$

Jack H.
Numerade Educator

Problem 28

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)

Jack H.
Numerade Educator

Problem 29

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)

Jack H.
Numerade Educator

Problem 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 $(\sqrt{2}, \sqrt{2})$

Jack H.
Numerade Educator

Problem 31

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

Jack H.
Numerade Educator

Problem 32

Find the area of one side of the "wall" standing perpendicularly 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 H.
Numerade Educator

Problem 33

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 H.
Numerade Educator

Problem 34

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 H.
Numerade Educator

Problem 35

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) $\delta=1$.

Jack H.
Numerade Educator

Problem 36

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

Jack H.
Numerade Educator

Problem 37

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 H.
Numerade Educator

Problem 38

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 yz-plane. Find the moments of inertia of the rod about the three coordinate axes.

Jack H.
Numerade Educator

Problem 39

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 $I_{2}$ 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.

Jack H.
Numerade Educator

Problem 40

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 $\bar{z}$ and $I_{z}$.

Jack H.
Numerade Educator

Problem 41

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

Jack H.
Numerade Educator

Problem 42

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

Jack H.
Numerade Educator

Problem 43

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.
$f(x, y, z)=\sqrt{1+30 x^{2}+10 y} ; \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+3 t^{2} \mathbf{k}$
$0 \leq t \leq 2$

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Problem 44

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.
$f(x, y, z)=\sqrt{1+x^{3}+5 y^{3}} ; \quad \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{2} \mathbf{j}+\sqrt{t} \mathbf{k}$
$0 \leq t \leq 2$

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Problem 45

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.
$f(x, y, z)=x \sqrt{y}-3 z^{2} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+5 t \mathbf{k}$
$0 \leq t \leq 2 \pi$

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Problem 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.
$f(x, y, z)=\left(1+\frac{9}{4} z^{1 / 3}\right)^{1 / 4} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+$
$t^{5 / 2} \mathbf{k}, \quad 0 \leq t \leq 2 \pi$

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