Calculus Early Transcendentals

James Stewart

Chapter 8

Further Applications of Integration - all with Video Answers

Educators

Section 1

Arc Length

Problem 1

Use the arc length formula (3) to find the length of the curve
$y=2 x-5,-1 \leqslant x \leqslant 3 .$ Check your answer by noting that
the curve is a line segment and calculating its length by the
distance formula.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 2

Use the arc length formula to find the length of the curve
$y=\sqrt{2-x^{2}}, 0 \leqslant x \leqslant 1 .$ Check your answer by noting that
the curve is part of a circle.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 3

$3-6$ Set up, but do not evaluate, an integral for the length of the
curve.
$$y=\cos x, \quad 0 \leqslant x \leqslant 2 \pi$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 4

$3-6$ Set up, but do not evaluate, an integral for the length of the
curve.
$$y=x e^{-x^{2}}, \quad 0 \leqslant x \leqslant 1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 5

$3-6$ Set up, but do not evaluate, an integral for the length of the
curve.
$$x=y+y^{3}, \quad 1 \leqslant y \leqslant 4$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 6

$3-6$ Set up, but do not evaluate, an integral for the length of the
curve.
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 7

$7-18$ Find the length of the curve.
$$y=1+6 x^{3 / 2}, \quad 0 \leqslant x \leqslant 1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 8

$7-18$ Find the length of the curve.
$$y^{2}=4(x+4)^{3}, \quad 0 \leqslant x \leqslant 2, \quad y>0$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 9

$7-18$ Find the length of the curve.
$$y=\frac{x^{5}}{6}+\frac{1}{10 x^{3}}, \quad 1 \leqslant x \leqslant 2$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 10

$7-18$ Find the length of the curve.
$$x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}, \quad 1 \leqslant y \leqslant 2$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 11

$7-18$ Find the length of the curve.
$$x=\frac{1}{3} \sqrt{y}(y-3), \quad 1 \leqslant y \leqslant 9$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 12

$7-18$ Find the length of the curve.
$$y=\ln (\cos x), \quad 0 \leqslant x \leqslant \pi / 3$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 13

$7-18$ Find the length of the curve.
$$y=\ln (\sec x), \quad 0 \leqslant x \leqslant \pi / 4$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 14

$7-18$ Find the length of the curve.
$$y=3+\frac{1}{2} \cosh 2 x, \quad 0 \leqslant x \leqslant 1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 15

$7-18$ Find the length of the curve.
$$y=\ln \left(1-x^{2}\right), \quad 0 \leqslant x \leqslant \frac{1}{2}$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 16

$7-18$ Find the length of the curve.
$$y=\sqrt{x-x^{2}}+\sin ^{-1}(\sqrt{x})$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 17

$7-18$ Find the length of the curve.
$$y=e^{x}, \quad 0 \leqslant x \leqslant 1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 18

$7-18$ Find the length of the curve.
$$y=\ln \left(\frac{e^{x}+1}{e^{x}-1}\right), \quad a \leqslant x \leqslant b, \quad a>0$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 19

$19-20$ Find the length of the arc of the curve from point $P$ to
point $Q .$
$$y=\frac{1}{2} x^{2}, \quad P\left(-1, \frac{1}{2}\right), \quad Q\left(1, \frac{1}{2}\right)$$

Chris Trentman

Numerade Educator

Problem 20

$19-20$ Find the length of the arc of the curve from point $P$ to
point $Q .$
$$x^{2}=(y-4)^{3}, \quad P(1,5), \quad Q(8,8)$$

Chris Trentman

Numerade Educator

Problem 21

$21-22$ Graph the curve and visually estimate its length. Then find
its exact length.
$$y=\frac{2}{3}\left(x^{2}-1\right)^{3 / 2}, \quad 1 \leqslant x \leqslant 3$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 22

$21-22$ Graph the curve and visually estimate its length. Then find
its exact length.
$$y=\frac{x^{3}}{6}+\frac{1}{2 x}, \quad \frac{1}{2} \leqslant x \leqslant 1$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 23

$23-26$ Use Simpson's Rule with $n=10$ to estimate the arc
length of the curve. Compare your answer with the value of the
integral produced by your calculator.
$$y=x e^{-x}, 0 \leqslant x \leqslant 5$$

Theresa Nguyen

Numerade Educator

Problem 24

$23-26$ Use Simpson's Rule with $n=10$ to estimate the arc
length of the curve. Compare your answer with the value of the
integral produced by your calculator.
$$x=y+\sqrt{y}, \quad 1 \leqslant y \leqslant 2$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 25

$23-26$ Use Simpson's Rule with $n=10$ to estimate the arc
length of the curve. Compare your answer with the value of the
integral produced by your calculator.
$$y=\sec x, \quad 0 \leqslant x \leqslant \pi / 3$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 26

$23-26$ Use Simpson's Rule with $n=10$ to estimate the arc
length of the curve. Compare your answer with the value of the
integral produced by your calculator.
$$y=x \ln x, \quad 1 \leqslant x \leqslant 3$$

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 27

(a) Graph the curve $y=x \sqrt[3]{4-x}, 0 \leqslant x \leqslant 4$
(b) Compute the lengths of inscribed polygons with $n=1,2$
and 4 sides. (Divide the interval into equal subintervals.)
Illustrate by sketching these polygons (as in Figure 6$)$(c) Set up an integral for the length of the curve.
(d) Use your calculator to find the length of the curve to four
decimal places. Compare with the approximations in
part (b).

AZ

Aaron Zsilavec

Numerade Educator

Problem 28

Repeat Exercise 27 for the curve
$$y=x+\sin x \quad 0 \leqslant x \leqslant 2 \pi$$

Chris Trentman

Numerade Educator

Problem 29

Use either a computer algebra system or a table of integrals to
find the exact length of the arc of the curve $y=\ln x$ that lies
between the points $(1,0)$ and $(2, \ln 2) .$

Chris Trentman

Numerade Educator

Problem 30

Use either a computer algebra system or a table of integrals to
find the exact length of the arc of the curve $y=x^{4 / 3}$ that lies
between the points $(0,0)$ and $(1,1) .$ If your CAS has trouble
evaluating the integral, make a substitution that changes the
integral into one that the CAS can evaluate.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 31

Sketch the curve with equation $x^{2 / 3}+y^{2 / 3}=1$ and use sym-
metry to find its length.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 32

(a) Sketch the curve $y^{3}=x^{2}$ .
(b) Use Formulas 3 and 4 to set up two integrals for the arc
length from $(0,0)$ to $(1,1) .$ Observe that one of these is
an improper integral and evaluate both of them.
(c) Find the length of the arc of this curve from (-1, 1)
to $(8,4) .$

Sam Low

Numerade Educator

Problem 33

Find the arc length function for the curve $y=2 x^{3 / 2}$ with
starting point $P_{0}(1,2) .$

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 34

(a) Graph the curve $y=\frac{1}{3} x^{3}+1 /(4 x), x>0$
(b) Find the arc length function for this curve with starting
point $P_{0}\left(1, \frac{7}{12}\right)$
(c) Graph the arc length function.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 35

Find the arc length function for the curve
$y=\sin ^{-1} x+\sqrt{1-x^{2}}$ with starting point $(0,1)$ .

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 36

A steady wind blows a kite due west. The kite's height above
ground from horizontal position $x=0$ to $x=80$ ft is given
by $y=150-\frac{1}{40}(x-50)^{2}$ . Find the distance traveled by the
kite.

Sean Veights

Numerade Educator

Problem 37

A hawk flying at 15 $\mathrm{m} / \mathrm{s}$ at an altitude of 180 $\mathrm{m}$ accidentally
drops its prey. The parabolic trajectory of the falling prey is
described by the equation
$$y=180-\frac{x^{2}}{45}$$
until it hits the ground, where $y$ is its height above the ground
and $x$ is the horizontal distance traveled in meters. Calculate
the distance traveled by the prey from the time it is dropped
until the time it hits the ground. Express your answer correct
to the nearest tenth of a meter.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 38

The Gateway Arch in St. Louis (see the photo on page 256$)$
was constructed using the equation
$$y=211.49-20.96 \cosh 0.03291765 x$$
for the central curve of the arch, where $x$ and $y$ are measured
in meters and $|x| \leqslant 91.20 .$ Set up an integral for the length
of the arch and use your calculator to estimate the length
correct to the nearest meter.

Chris Trentman

Numerade Educator

Problem 39

A manufacturer of corrugated metal roofing wants to produce
panels that are 28 in. wide and 2 in. thick by processing flat
sheets of metal as shown in the figure. The profile of the roof-
ing takes the shape of a sine wave. Verify that the sine curve
has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat
metal sheet that is needed to make a 28 -inch panel. (Use your
calculator to evaluate the integral correct to four significant
digits.)

Clarissa Noh

Numerade Educator

Problem 40

(a) The figure shows a telephone wire hanging between
two poles at $x=-b$ and $x=b .$ It takes the shape of a
catenary with equation $y=c+a \cosh (x / a) .$ Find the
length of the wire.
(b) Suppose two telephone poles are 50 $\mathrm{ft}$ apart and the
length of the wire between the poles is 51 $\mathrm{ft}$ . If the lowest
point of the wire must be 20 $\mathrm{ft}$ above the ground, how
high up on each pole should the wire be attached?

Linda Hand

Numerade Educator

Problem 41

Find the length of the curve
$$y=\int_{1}^{x} \sqrt{t^{3}-1} d t \quad 1 \leqslant x \leqslant 4$$

Chris Trentman

Numerade Educator

Problem 42

The curves with equations $x^{n}+y^{n}=1, n=4,6,8, \ldots,$ are
called fat circles. Graph the curves with $n=2,4,6,8,$ and
10 to see why. Set up an integral for the length $L_{2 k}$ of the fat
circle with $n=2 k$ . Without attempting to evaluate this inte-
gral, state the value of $\lim _{k \rightarrow \infty} L_{2 k}.$

Chris Trentman

Numerade Educator