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Chapter 16

Vector Calculus

Educators


Problem 1

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C y \, ds $, $ C: x = t^2 $, $ y = 2t $, $ 0 \leqslant t \leqslant 3 $

Frank L.
Numerade Educator

Problem 2

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C (x/y) \, ds $, $ C: x = t^3 $, $ y = t^4 $, $ 1 \leqslant t \leqslant 2 $

Frank L.
Numerade Educator

Problem 3

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C xy^4 \, ds $, $ C $ is the right half of the circle $ x^2 + y^2 = 16 $

Frank L.
Numerade Educator

Problem 4

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C xe^y \, ds $, $ C $ is the line segment from $ (2, 0) $ to $ (5, 4) $

Frank L.
Numerade Educator

Problem 5

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C (x^2y + \sin x) \, dy $,
$ C $ is the arc of the parabola $ y = x^2 $ from $ (0, 0) $ to $ (\pi, \pi^2) $.

Frank L.
Numerade Educator

Problem 6

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C e^x \, dx $,
$ C $ is the arc of the curve $ x = y^3 $ from $ (-1, -1) $ to $ (1, 1) $

Frank L.
Numerade Educator

Problem 7

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C (x+ 2y) \, dx + x^2 \, dy $, $ C $ consists of line segments from $ (0, 0) $ to $ (2, 1) $ and from $ (2, 1) $ to $ (3, 0) $

Frank L.
Numerade Educator

Problem 8

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C x^2 \, dx + y^2 \, dy $, $ C $ consists of the arc of the circle $ x^2 + y^2 = 4 $ from $ (2, 0) $ to $ (0, 2) $ followed by the line segment from $ (0, 2) $ to $ (4, 3) $

Frank L.
Numerade Educator

Problem 9

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C x^2y \, ds $,
$ C: x = \cos t $, $ y = \sin t $, $ z = t $, $ 0 \leqslant t \leqslant \pi/2 $

Frank L.
Numerade Educator

Problem 10

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C y^2z \, ds $,
$ C $ is the line segment from $ (3, 1, 2) $ to $(1, 2, 5) $

Frank L.
Numerade Educator

Problem 11

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C xe^{yz} \, ds $,
$ C $ is the line segment from $ (0, 0, 0) $ to $ (1, 2, 3) $

Frank L.
Numerade Educator

Problem 12

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C (x^2 + y^2 + z^2) \, ds $,
$ C: x = t $, $ y = \cos 2t $, $ z = \sin 2t $, $ 0 \leqslant t \leqslant 2\pi $

Bobby B.
University of North Texas

Problem 13

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C xye^{yz} \, dy $,
$ C: x = t $, $ y = t^2 $, $ z = t^3 $, $ 0 \leqslant t \leqslant 1 $

Linda H.
Numerade Educator

Problem 14

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C y \, dx + z \, dy + x \, dz $,
$ C: x = \sqrt{t} $, $ y = t $, $ z = t^2 $, $ 1 \leqslant t \leqslant 4 $

Frank L.
Numerade Educator

Problem 15

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C z^2 \, dx + x^2 \, dy + y^2 \, dz $,
$ C $ is the line segment from $ (1, 0, 0) $ to $ (4, 1, 2) $

Frank L.
Numerade Educator

Problem 16

Evaluate the line integral, where $ C $ is the given curve.

$ \displaystyle \int_C ( y + z) \, dx + (x + z) \, dy + (x + y) \, dz $,
$ C $ consists of line segments from $ (0, 0, 0) $ to $ (1, 0, 1) $ and from $ (1, 0, 1) $ to $ (0, 1, 2) $

Frank L.
Numerade Educator

Problem 17

Let $ \textbf{F} $ be the vector field shown in the figure.
(a) If $ C_1 $ is the vertical line segment from $ (-3, -3) $ to $ (-3, 3) $, determine whether $ \displaystyle \int_{C_1} \textbf{F} \cdot d \textbf{r} $ is positive, negative, or zero.
(b) If $ C_2 $ is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether $ \displaystyle \int_{C_2} \textbf{F} \cdot d \textbf{r} $ is positive, negative, or zero.

Frank L.
Numerade Educator

Problem 18

The figure shows a vector field $ \textbf{F} $ and two curves $ C_1 $ and $ C_2 $. Are the line integrals of $ \textbf{F} $ over $ C_1 $ and $ C_2 $ positive, negative, or zero? Explain.

Frank L.
Numerade Educator

Problem 19

Evaluate the line integral $ \displaystyle \int_C \textbf{F} \cdot d \textbf{r} $, where $ C $ is given by the vector function $ \textbf{r} (t) $.

$ \textbf{F}(x, y) = xy^2 \textbf{i} - x^2 \textbf{j} $,
$ \textbf{r}(t) = t^3 \textbf{i} + t^2 \textbf{j} $, $ 0 \leqslant t \leqslant 1 $

Frank L.
Numerade Educator

Problem 20

Evaluate the line integral $ \displaystyle \int_C \textbf{F} \cdot d \textbf{r} $, where $ C $ is given by the vector function $ \textbf{r} (t) $.

$ \textbf{F}(x, y, z) = (x + y^2) \textbf{i} + xz \textbf{j} + (y + z) \textbf{k} $,
$ \textbf{r}(t) = t^2 \textbf{i} + t^3 \textbf{j} - 2t \textbf{k} $, $ 0 \leqslant t \leqslant 2 $

Frank L.
Numerade Educator

Problem 21

Evaluate the line integral $ \displaystyle \int_C \textbf{F} \cdot d \textbf{r} $, where $ C $ is given by the vector function $ \textbf{r} (t) $.

$ \textbf{F}(x, y, z) = \sin x \textbf{i} + \cos y \textbf{j} + xz \textbf{k} $,
$ \textbf{r}(t) = t^3 \textbf{i} - t^2 \textbf{j} + t \textbf{k} $, $ 0 \leqslant t \leqslant 1 $

Frank L.
Numerade Educator

Problem 22

Evaluate the line integral $ \displaystyle \int_C \textbf{F} \cdot d \textbf{r} $, where $ C $ is given by the vector function $ \textbf{r} (t) $.

$ \textbf{F}(x, y, z) = x \textbf{i} + y \textbf{j} + xy \textbf{k} $,
$ \textbf{r}(t) = \cos t \textbf{i} + \sin t \textbf{j} + t \textbf{k} $, $ 0 \leqslant t \leqslant \pi $

Frank L.
Numerade Educator

Problem 23

Use a calculator to evaluate the line integral correct to four decimal places.

$ \displaystyle \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y) = \sqrt{x + y} \, \textbf{i} + (y/x) \, \textbf{j} $ and $ \textbf{r}(t) = \sin^2 t \, \textbf{i} + \sin t \cos t \, \textbf{j} $, $ \pi/6 \leqslant t \leqslant \pi/3 $

Frank L.
Numerade Educator

Problem 24

Use a calculator to evaluate the line integral correct to four decimal places.

$ \displaystyle \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y, z) = yze^x \, \textbf{i} + zxe^y \, \textbf{j} + xye^z \, \textbf{k}$ and $ \textbf{r}(t) = \sin t \, \textbf{i} + \cos t \, \textbf{j} + \tan t \, \textbf{k} $, $ 0 \leqslant t \leqslant \pi/4 $

Frank L.
Numerade Educator

Problem 25

Use a calculator to evaluate the line integral correct to four decimal places.

$ \displaystyle \int_C xy \arctan z \, ds $, where $ C $ has parametric equations $ x = t^2 $, $ y = t^3 $, $ z = \sqrt{t} $, $ 1 \leqslant t \leqslant 2 $

Frank L.
Numerade Educator

Problem 26

Use a calculator to evaluate the line integral correct to four decimal places.

$ \displaystyle \int_C z \ln (x + y) \, ds $, where $ C $ has parametric equations $ x = 1 + 3t $, $ y = 2 + t^2 $, $ z = t^4 $, $ -1 \leqslant t \leqslant 1 $

Frank L.
Numerade Educator

Problem 27

Use a graph of the vector field $ \textbf{F} $ and the curve $ C $ to guess whether the line integral of $ \textbf{F} $ over $ C $ is positive, negative, or zero. Then evaluate the line integral.

$ \textbf{F}(x, y) = (x - y) \, \textbf{i} + xy \, \textbf{j} $,
$ C $ is the arc of the circle $ x^2 + y^2 = 4 $ traversed counter-clockwise from $ (2, 0) $ to $ (0, -2) $

Frank L.
Numerade Educator

Problem 28

Use a graph of the vector field $ \textbf{F} $ and the curve $ C $ to guess whether the line integral of $ \textbf{F} $ over $ C $ is positive, negative, or zero. Then evaluate the line integral.

$ \textbf{F}(x, y) = \dfrac{x}{\sqrt{x^2 + y^2}} \, \textbf{i} + \dfrac{y}{\sqrt{x^2 + y^2}} \, \textbf{j} $,
$ C $ is the parabola $ y = 1 + x^2 $ from $ (-1, 2) $ to $ (1, 2) $

Frank L.
Numerade Educator

Problem 29

(a) Evaluate the line integral $ \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y) = e^{x - 1} \, \textbf{i} + xy \, \textbf{j} $ and $ C $ is given by $ \textbf{r}(t) = t^2 \, \textbf{i} + t^3 \, \textbf{j} $, $ 0 \leqslant t \leqslant 1 $.

(b) Illustrate part (a) by using a graphing calculator or computer to graph $ C $ and the vectors from the vector field corresponding to $ t = 0, 1/\sqrt{2} $, and $ 1 $ (as in Figure 13).

Frank L.
Numerade Educator

Problem 30

(a) Evaluate the line integral $ \int_C \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F}(x, y, z) = x
\, \textbf{i} - z \, \textbf{j} + y \, \textbf{k} $ and $ C $ is given by $ \textbf{r}(t) = 2t \, \textbf{i} + 3t \, \textbf{j} - t^2 \, \textbf{k} $, $ -1 \leqslant t \leqslant 1 $.

(b) Illustrate part (a) by using a computer to graph $ C $ and the vectors from the vector field corresponding to $ t = \pm 1 $ and $ \pm \frac{1}{2} $ (as in Figure 13).

Frank L.
Numerade Educator

Problem 31

Find the exact value of $ \int_C x^3y^2z \, ds $, where $ C $ is the curve with parametric equations $ x = e^{-t} \cos 4t $, $ y = e^{-t} \sin 4t $, $ z = e^{-t} $, $ 0 \leqslant t \leqslant 2 \pi $.

Frank L.
Numerade Educator

Problem 32

(a) Find the work done by the force field $ \textbf{F} (x, y) = x^2 \, \textbf{i} + xy \, \textbf{j} $ on a particle that moves once around the circle $ x^2 + y^2 = 4 $ oriented in the counter-clockwise direction.
(b) Use a computer algebra system to graph the force field and circle on the same screen. Use the graph to explain your answer to part (a).

Frank L.
Numerade Educator

Problem 33

A thin wire is bent into the shape of a semicircle $ x^2 + y^2 = 4 $, $ x \geqslant 0 $. If the linear density is a constant $ k $, find the mass and center of mass of the wire.

Frank L.
Numerade Educator

Problem 34

A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius $ a $. If the density function is $ \rho (x, y) = kxy $, find the mass and center of mass of the wire.

Frank L.
Numerade Educator

Problem 35

(a) Write the formulas similar to Equations 4 for the center of mass $ (\bar{x}, \bar{y}, \bar{z}) $ of a thin wire in the shape of a space curve $ C $ if the wire has density function $ \rho(x, y, z) $.

(b) Find the center of mass of a wire in the shape of the helix $ x = 2 \sin t $, $ y = 2 \cos t $, $ z = 3t $, $ 0 \leqslant t \leqslant 2\pi $, if the density is a constant $ k $.

Frank L.
Numerade Educator

Problem 36

Find the mass and center of mass of a wire in the shape of the helix $ x = t $, $ y = \cos t $, $ z = \sin t $, $ 0 \leqslant t \leqslant 2\pi $, if the density at any point is equal to the square of the distance from the origin.

Frank L.
Numerade Educator

Problem 37

If a wire with linear density $ \rho(x, y) $ lies along a plane curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $- and $ y $-axes are defined as
$ I_x = \int_C y^2 \rho(x, y) ds $ $ I_y = \int_C x^2 \rho(x, y) ds $

Find the moments of inertia for the wire in Example 3.

Frank L.
Numerade Educator

Problem 38

If a wire with linear density $ \rho(x, y, z) $ lies along a space curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $-, $ y $-, and $ z $-axes are defined as
$$ I_x = \int_C (y^2 + z^2) \rho(x, y, z) ds $$
$$ I_y = \int_C (x^2 + z^2) \rho(x, y, z) ds $$
$$ I_z = \int_C (x^2 + y^2) \rho(x, y, z) ds $$

Find the moments of inertia for the wire in Exercise 35.

Frank L.
Numerade Educator

Problem 39

Find the work done by the force field $$ \textbf{F}(x, y) = x \, \textbf{i} + (y + 2) \, \textbf{j} $$ in moving an object along an arch of the cycloid
$ \textbf{r}(t) = (t - \sin t) \, \textbf{i} + (1 - \cos t) \, \textbf{j} $ $ 0 \leqslant t \leqslant 2\pi $

Frank L.
Numerade Educator

Problem 40

Find the work done by the force field $ \textbf{F}(x, y) = x^2 \, \textbf{i} + ye^x \, \textbf{j} $ on a particle that moves along the parabola $ x = y^2 + 1 $ from $ (1, 0) $ to $ (2, 1) $.

Frank L.
Numerade Educator

Problem 41

Find the work done by the force field $$ \textbf{F}(x, y, z) = \langle x - y^2, y - z^2, z - x^2 \rangle $$ on a particle that moves along the line segment from $ (0, 0, 1) $ to $ (2, 1, 0) $.

Frank L.
Numerade Educator

Problem 42

The force exerted by an electric charge at the origin on a charged particle at a point $ (x, y, z) $ with position vector $ \textbf{r} = \langle x, y, z \rangle $ is $ \textbf{F(r)} = K \textbf{r} / | \textbf{r} |^3 $ where $ K $ is a constant. (See Example 16.1.5) Find the work done as the particle moves along a straight line from $ (2, 0, 0) $ to $ (2, 1, 5) $.

Frank L.
Numerade Educator

Problem 43

The position of an object with mass $ m $ at time $ t $ is $ \textbf{r}(t) = at^2 \, \textbf{i} + bt^3 \, \textbf{j} $, $ 0 \leqslant t \leqslant 1 $.

(a) What is the force acting on the object at time $ t $?
(b) What is the work done by the force during the time interval $ 0 \leqslant t \leqslant 1 $?

Frank L.
Numerade Educator

Problem 44

An object with mass $ m $ moves with position function $ \textbf{r}(t) = a \sin t \, \textbf{i} + b \cos t \, \textbf{j} + ct \textbf{k} $, $ 0 \leqslant t \leqslant \pi/2 $. Find the work done on the object during this time period.

Frank L.
Numerade Educator

Problem 45

A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?

Frank L.
Numerade Educator

Problem 46

Suppose there is a hole in the can of paint in Exercise 45 and 9 lb of paint leaks steadily out of the can during the man's ascent. How much work is done?

Carson M.
Numerade Educator

Problem 47

(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle $ x^2 + y^2 = 1 $.

(b) Is this also true for a force field $ \textbf{F(x)} = k\textbf{x} $, where $ k $ is a constant and $ \textbf{x} = \langle x, y \rangle $?

Frank L.
Numerade Educator

Problem 48

The base of a circular fence with radius 10 m is given by $ x = 10 \cos t $, $ y = 10 \sin t $. The height of the fence at position $ (x, y) $ is given by the function $ h(x, y) = 4 + 0.01(x^2 - y^2) $, so the height varies from $ 3 m $ to $ 5 m $. Suppose that $ 1 L $ of paint covers $ 100 m^2 $. Sketch the fence and determine how much paint you will need if you paint both sides of the fence.

Frank L.
Numerade Educator

Problem 49

If $ C $ is a smooth curve given by a vector function $ \textbf{r}(t) $, $ a \leqslant t \leqslant b $, and $ \textbf{v} $ is a constant vector, show that $$ \int_C \textbf{v} \cdot d \textbf{r} = \textbf{v} \cdot [ \textbf{r}(b) - \textbf{r}(a)] $$

Frank L.
Numerade Educator

Problem 50

If $ C $ is a smooth curve given by a vector function $ \textbf{r}(t) $, $ a \leqslant t \leqslant b $, show that $$ \int_C \textbf{r} \cdot d \textbf{r} = \frac{1}{2} \bigl[ | \textbf{r}(b) |^2 - | \textbf{r}(a) |^2 \bigr] $$

Frank L.
Numerade Educator

Problem 51

An object moves along the curve $ C $ shown in the figure from $ (1, 2) $ to $ (9, 8) $. The lengths of the vectors in the force field $ \textbf{F} $ are measured in newtons by the scales on the axes. Estimate the work done by $ \textbf{F} $ on the object.

Frank L.
Numerade Educator

Problem 52

Experiments show that a steady current $ I $ in a long wire produces a magnetic field $ \textbf{B} $ that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire (as in the figure). $ \textit{Ampere's Law} $ relates the electric

Frank L.
Numerade Educator