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 an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
$$y=\sin x, \quad 0 \leqslant x \leqslant \pi$$

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Problem 4

$3-6$ Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
$$y=x e^{-x}, 0 \leqslant x \leqslant 2$$

Chris Trentman

Numerade Educator

Problem 5

$3-6$ Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
$$x=\sqrt{y}-y, 1 \leqslant y \leqslant 4$$

Chris Trentman

Numerade Educator

Problem 6

$3-6$ Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
$$x=y^{2}-2 y, \quad 0 \leqslant y \leqslant 2$$

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Numerade Educator

Problem 7

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

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Problem 8

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

Chris Trentman

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Problem 9

$7-18$ Find the exact length of the curve.
$$y=\frac{x^{3}}{3}+\frac{1}{4 x}, 1 \leq x \leq 2$$

Chris Trentman

Numerade Educator

Problem 10

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

Chris Trentman

Numerade Educator

Problem 11

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

Chris Trentman

Numerade Educator

Problem 12

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

Chris Trentman

Numerade Educator

Problem 13

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

Chris Trentman

Numerade Educator

Problem 14

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

Chris Trentman

Numerade Educator

Problem 15

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

Chris Trentman

Numerade Educator

Problem 16

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

Chris Trentman

Numerade Educator

Problem 17

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

Chris Trentman

Numerade Educator

Problem 18

$7-18$ Find the exact length of the curve.
$$y=1-e^{-x}, \quad 0 \leqslant x \leqslant 2$$

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Numerade Educator

Problem 19

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

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Problem 20

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

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Numerade Educator

Problem 21

$21-22$ Graph the curve and visually estimate its length. Then use your calculator to find the length correct to four decimal places.
$$y=x^{2}+x^{3}, \quad 1 \leqslant x \leqslant 2$$

Chris Trentman

Numerade Educator

Problem 22

$21-22$ Graph the curve and visually estimate its length. Then use your calculator to find the length correct to four decimal places.
$$y=x+\cos x, \quad 0 \leqslant x \leqslant \pi / 2$$

Chris Trentman

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.
$$y=x \sin x, \quad 0 \leqslant x \leqslant 2 \pi$$

Chris Trentman

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.
$$y=\sqrt[3]{x}, \quad 1 \leqslant x \leqslant 6$$

Chris Trentman

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.
$$y=\ln \left(1+x^{3}\right), \quad 0 \leqslant x \leqslant 5$$

Chris Trentman

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.
$$y=e^{-x^{2}}, \quad 0 \leqslant x \leqslant 2$$

Chris Trentman

Numerade Educator

Problem 27

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

Chris Trentman

Numerade Educator

Problem 28

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

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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
symmetry to find its length.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 32

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

Chris Trentman

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

$\begin{array}{l}{\text { (a) Find the arc length function for the curve } y=\ln (\sin x)} \\ {0 < x < \pi, \text { with starting point }(\pi / 2,0) .} \\ {\text { (b) Graph both the curve and its arc length function on the }} \\ {\text { same screen. }}\end{array}$

Chris Trentman

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.

Rakesh Kumar Sharma

Rakesh Kumar Sharma

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 259)
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 produc
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
roofing 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 signifi-
cant digits.)

Chris Trentman

Numerade Educator

Problem 40

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

Chris Trentman

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