Calculus: Early Transcendentals, Metric Edition

James Stewart, Daniel K. Clegg, Saleem Watson

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 $y=3-2 x,-1 \leqslant x \leqslant 3 .$ Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Use the arc length formula to find the length of the curve $y=\sqrt{4-x^{2}}, 0 \leqslant x \leqslant 2$. Check your answer by noting that the curve is part of a circle.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 3

3-8 Set up, but do not evaluate, an integral for the length of the curve.
$$
y=x^{3}, \quad 0 \leqslant x \leqslant 2
$$

Nick Johnson

Numerade Educator

Problem 4

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

Nick Johnson

Numerade Educator

Problem 5

Set up, but do not evaluate, an integral for the length of the curve.
$$
y=x-\ln x, \quad 1 \leqslant x \leqslant 4
$$

Nick Johnson

Numerade Educator

Problem 6

Set up, but do not evaluate, an integral for the length of the curve.
$$
x=y^{2}+y, \quad 0 \leq y \leq 3
$$

Nick Johnson

Numerade Educator

Problem 7

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

Nick Johnson

Numerade Educator

Problem 8

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

Samantha Lail

Numerade Educator

Problem 9

9-24 Find the exact length of the curve.
$y=\frac{2}{3} x^{3 / 2}, \quad 0 \leq x \leq 2$

Gregory Higby

Numerade Educator

Problem 10

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

Samantha Lail

Numerade Educator

Problem 11

Find the exact length of the curve.
$$
y=\frac{2}{3}\left(1+x^{2}\right)^{3 / 2}, \quad 0 \leq x \leq 1
$$

Gregory Higby

Numerade Educator

Problem 12

Find the exact length of the curve.
$$
36 y^{2}=\left(x^{2}-4\right)^{3}, \quad 2 \leq x \leq 3, \quad y \geqslant 0
$$

Gregory Higby

Numerade Educator

Problem 13

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 14

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 15

Find the exact length of the curve.
$$
y=\frac{1}{2} \ln (\sin 2 x), \quad \pi / 8 \leq x \leq \pi / 6
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 16

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

Nick Johnson

Numerade Educator

Problem 17

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 18

Find the exact length of the curve.
$$
x=e^{y}+\frac{1}{4} e^{-y}, \quad 0 \leqslant y \leq 1
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 19

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 20

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 21

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 22

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

Clarissa Noh

Numerade Educator

Problem 23

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 24

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 25

25-26 Find the length of the arc of the curve from point $P$ to point $Q$.
$$
y=\frac{1}{2} x^{2}, \quad P\left(-1, \frac{1}{2}\right), \quad Q\left(1, \frac{1}{2}\right)
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 26

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 27

27-32 Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places.
$$
y=x^{2}+x^{3}, \quad 1 \leqslant x \leqslant 2
$$

Gregory Higby

Numerade Educator

Problem 28

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

Gregory Higby

Numerade Educator

Problem 29

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places.
$$
y=\sqrt[3]{x}, \quad 1 \leqslant x \leqslant 4
$$

Gregory Higby

Numerade Educator

Problem 30

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places.
$$
y=x \tan x, \quad 0 \leqslant x \leqslant 1
$$

Gregory Higby

Numerade Educator

Problem 31

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places.
$$
y-x e^{-x}, \quad 1 \leq x \leqslant 2
$$

Gregory Higby

Numerade Educator

Problem 32

Graph the curve and visually estimate its length. Then compute the length, correct to four decimal places.
$$
y=\ln \left(x^{2}+4\right), \quad-2 \leqslant x \leq 2
$$

Gregory Higby

Numerade Educator

Problem 33

33-34 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 a calculator or computer.
$$
y=x \sin x, 0 \leqslant x \leqslant 2 \pi
$$

James Kiss

Numerade Educator

Problem 34

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 a calculator or computer.
$$
y=e^{-x^{2}}, 0 \leqslant x \leqslant 2
$$

James Kiss

Numerade Educator

Problem 35

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

James Kiss

Numerade Educator

Problem 36

Repeat Exercise 35 for the curve
$$
y=x+\sin x \quad 0 \leqslant x \leqslant 2 \pi
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 37

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

James Kiss

Numerade Educator

Problem 38

Use either a computer 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 software has trouble evaluating the integral, make a substitution that changes the integral into one that the software can evaluate.

James Kiss

Numerade Educator

Problem 39

$$
\text { Find the length of the astroid } x^{2 / 3}+y^{2 / 3}=1 \text { . }
$$

James Kiss

Numerade Educator

Problem 40

(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 $(8,4)$.

Chris Trentman

Numerade Educator

Problem 41

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 42

(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. Why is the arc length function negative when $x$ is less than $\pi / 2$ ?

James Kiss

Numerade Educator

Problem 43

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 44

The arc length function for a curve $y=f(x)$, where $f$ is an increasing function, is $s(x)=\int_{0}^{x} \sqrt{3 t+5} d t$.
(a) If $f$ has $y$ -intercept 2 , find an equation for $f$.
(b) What point on the graph of $f$ is 3 units along the curve from the $y$ -intercept? State your answer rounded to 3 decimal places.

Anthony Ramos

Numerade Educator

Problem 45

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.

Clarissa Noh

Numerade Educator

Problem 46

A steady wind blows a kite due west. The kite's height above ground from horizontal position $x=0$ to $x=25 \mathrm{~m}$ is given by $y=50-0.1(x-15)^{2}$. Find the distance traveled by the kite.

James Kiss

Numerade Educator

Problem 47

A manufacturer of corrugated metal roofing wants to produc panels that are $60 \mathrm{~cm}$ wide and $4 \mathrm{~cm}$ high 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 $y=2 \sin (\pi x / 15)$ and find the width $w$ of a flat metal sheet that is needed to make a $60-\mathrm{cm}$ panel. (Numerically evaluate the integral correct to four significant digits.)

Clarissa Noh

Numerade Educator

Problem 48

50 Catenary Curves A chain (or cable) of uniform density that is suspended between two points, as shown in the figure, hangs in the shape of a curve called a catenary with equation $y=a \cosh (x / a) .
(a) Find the arc length of the catenary $y=a \cosh (x / a)$ on the interval $[c, d]$.
(b) Show that on any interval $[c, d]$, the ratio of the area under the catenary to its arc length is $a$.

James Kiss

Numerade Educator

Problem 49

Catenary Curves A chain (or cable) of uniform density that is suspended between two points, as shown in the figure, hangs in the shape of a curve called a catenary with equation $y=a \cosh (x / a) .
The figure shows a telephone wire hanging between two poles at $x=-10$ and $x=10$. The wire hangs in the shape of a catenary described by the equation
$$
y=c+a \cosh \frac{x}{a}
$$
If the length of the wire between the two poles is $20.4 \mathrm{~m}$ and the lowest point of the wire must be $9 \mathrm{~m}$ above the ground, how high up on each pole should the wire be attached?

James Kiss

Numerade Educator

Problem 50

Catenary Curves A chain (or cable) of uniform density that is suspended between two points, as shown in the figure, hangs in the shape of a curve called a catenary with equation $y=a \cosh (x / a) .
50. The British physicist and architect Robert Hooke (16351703 ) was the first to observe that the ideal shape for a free standing arch is an inverted catenary. Hooke remarked, "As hangs the chain, so stands the arch." The Gateway Arch in St. Louis is based on the shape of a catenary; the central curve of the arch is modeled by the equation
$$
y=211.49-20.96 \cosh 0.03291765 x
$$
where $x$ and $y$ are measured in meters and $|x| \leqslant 91.20$. Set up an integral for the length of the arch and evaluate the integral numerically to estimate the length correct to the nearest meter.

James Kiss

Numerade Educator

Problem 51

For the function $f(x)=\frac{1}{4} e^{x}+e^{-x}$, prove that the arc length on any interval has the same value as the area under the
curve.

Clarissa Noh

Numerade Educator

Problem 52

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 $\lim _{k \rightarrow \infty} L_{2 k-}$

Christian Otero

Christian Otero

Numerade Educator

Problem 53

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