Essential Calculus Early Transcendentals

James Stewart

Chapter 7

APPLICATIONS OF INTEGRATION - all with Video Answers

Educators

Section 4

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 S.

Rakesh Kumar S.

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 S.

Rakesh Kumar S.

Numerade Educator

Problem 3

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 4

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 5

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 7

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 8

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 9

Find the exact length of the curve.
$$y=\frac{x^{3}}{3}+\frac{1}{4 x}, \quad 1 \leqslant x \leqslant 2$$

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 10

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 11

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 12

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 13

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 14

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 15

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 16

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 17

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 18

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 19

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 20

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 21

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 22

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 23

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 24

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 25

Use either a computer algebra system or a table of
integrals to find the exact length of the arc of the curve
$x=\ln \left(1-y^{2}\right)$ that lies between the points $(0,0)$ and
$\left(\ln \frac{3}{4}, \frac{1}{2}\right)$

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 26

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 S.

Rakesh Kumar S.

Numerade Educator

Problem 27

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 28

(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 $(8,4) .$

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Problem 29

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 30

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

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Problem 31

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

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 32

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 S.

Rakesh Kumar S.

Numerade Educator

Problem 33

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 S.

Rakesh Kumar S.

Numerade Educator

Problem 34

(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 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?

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Problem 35

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 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 significant digits.)

Rakesh Kumar S.

Rakesh Kumar S.

Numerade Educator

Problem 36

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 integral, state the value of $\lim _{k \rightarrow \infty} L_{2 k}$ .

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