Calculus

James Stewart

Chapter 8

Further Applications of Integration - all with Video Answers

Educators

Section 1

8.1 Exercises

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

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 12

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

Nick Johnson

Numerade Educator

Problem 13

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 14

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

Clarissa Noh

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 16

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 18

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 19

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

Clarissa Noh

Numerade Educator

Problem 20

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 21

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 22

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 25

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 Sharma

Rakesh Kumar Sharma

Numerade Educator

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

Rakesh Kumar Sharma

Rakesh Kumar Sharma

Numerade Educator

Problem 27

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.
(a) Graph the curve $y=x \sqrt[3]{4-x}, 0 \leq x \leq 4$
(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).

Carson Merrill

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 symmetry to find its length.

Clarissa Noh

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 are 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) 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.

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 $(0,1)$.

Clarissa Noh

Numerade Educator

Problem 36

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

Clarissa Noh

Numerade Educator

Problem 37

A hawk flying at $15 \mathrm{~m} / \mathrm{s}$ at an altitude of 180 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 463) 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.

Clarissa Noh

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 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 Sharma

Rakesh Kumar Sharma

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 ft apart and the length of the wire between the poles is 51 ft . If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?

Clarissa Noh

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

Christian Otero

Christian Otero

Numerade Educator