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Chapter 8

Further Applications of Integration

Educators

HO

Problem 1

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

Clarissa N.
Numerade Educator

Problem 2

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

HO
Harriet O.
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 $, $ 0 \le x \le \pi $

Clarissa N.
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 = xe^{-x} $, $ 0 \le x \le 2 $

HO
Harriet O.
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.

$ y = x - \ln x $, $ 1 \le x \le 4 $

Clarissa N.
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 - 2y $, $ 0 \le y \le 2 $

HO
Harriet O.
Numerade Educator

Problem 7

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 \le y \le 4 $

Clarissa N.
Numerade Educator

Problem 8

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^2 = \ln x $, $ -1 \le y \le 1 $

HO
Harriet O.
Numerade Educator

Problem 9

Find the exact length of the curve.

$ y = 1 + 6x^{\frac{3}{2}} $ , $ 0 \le x \le 1 $

Clarissa N.
Numerade Educator

Problem 10

Find the exact length of the curve.

$ 36y^2 = (x^2 - 4)^3 $ , $ 2 \le x \le 3 $ , $ y \ge 0 $

HO
Harriet O.
Numerade Educator

Problem 11

Find the exact length of the curve.

$ y = \displaystyle \frac{x^3}{3} + \displaystyle \frac{1}{4x} $ , $ 1 \le x \le 2 $

Clarissa N.
Numerade Educator

Problem 12

Find the exact length of the curve.

$ x = \displaystyle \frac{y^4}{8} + \displaystyle \frac{1}{4y^2} $ , $ 1 \le y \le 2 $

Carson M.
Numerade Educator

Problem 13

Find the exact length of the curve.

$ x = \frac{1}{3} \sqrt{y} (y - 3) $ , $ 1 \le y \le 9 $

Clarissa N.
Numerade Educator

Problem 14

Find the exact length of the curve.

$ y = \ln (\cos x) $ , $ 0 \le x \le \frac{\pi}{3} $

HO
Harriet O.
Numerade Educator

Problem 15

Find the exact length of the curve.

$ y = \ln (\sec x) $ , $ 0 \le x \le \frac{\pi}{4} $

Clarissa N.
Numerade Educator

Problem 16

Find the exact length of the curve.

$ y = 3 + \frac{1}{2} \cosh 2x $ , $ 0 \le x \le 1 $

Clarissa N.
Numerade Educator

Problem 17

Find the exact length of the curve.

$ y = \frac{1}{4} x^2 - \frac{1}{2} \ln x $ , $ 1 \le x \le 2 $

Clarissa N.
Numerade Educator

Problem 18

Find the exact length of the curve.

$ y = \sqrt{x - x^2} + \sin^{-1} (\sqrt{x}) $

Clarissa N.
Numerade Educator

Problem 19

Find the exact length of the curve.

$ y = \ln (1 - x^2) $ , $ 0 \le x \le \frac{1}{2} $

Clarissa N.
Numerade Educator

Problem 20

Find the exact length of the curve.

$ y = 1 - e^{-x} $ , $ 0 \le x \le 2 $

Clarissa N.
Numerade Educator

Problem 21

Find the length of the arc of the curve from point $ P $ to point $ Q $.

$ y = \frac{1}{2} x^2 $ , $ P(-1, \frac{1}{2}) $ , $ Q(1, \frac{1}{2}) $

Clarissa N.
Numerade Educator

Problem 22

Find the length of the arc of the curve from point $ P $ to point $ Q $.

$ x^2 = (y - 4)^3 $ , $ P(1, 5) $ , $ Q(8, 8) $

Clarissa N.
Numerade Educator

Problem 23

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 $ , $ 1 \le x \le 2 $

Clarissa N.
Numerade Educator

Problem 24

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 $ , $ 0 \le x \le \frac{\pi}{2} $

Clarissa N.
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 a calculator.

$ y = x \sin x $ , $ 0 \le x \le 2\pi $

Clarissa N.
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 a calculator.

$ y = \sqrt[3]{x} $ , $ 1 \le x \le 6 $

Clarissa N.
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 a calculator.

$ y = \ln (1 + x^3) $ , $ 0 \le x \le 5 $

Clarissa N.
Numerade Educator

Problem 28

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.

$ y = e^{-x^2} $ , $ 0 \le x \le 2 $

Clarissa N.
Numerade Educator

Problem 29

(a) Graph the curve $ y = x \sqrt[3]{4 - x} $ , $ 0 \le x \le 4 $.
(b) Compute the lengths of inscribed polygons with $ n = 1 $, 2, and 4 sides. (Divide the interval into equal sub-intervals.) 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).

Clarissa N.
Numerade Educator

Problem 30

Repeat Exercise 29 for the curve
$ y = x + \sin x $ $ 0 \le x \le 2\pi $

Clarissa N.
Numerade Educator

Problem 31

Use either a computer algebra system 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 $ (2, e^2) $.

Clarissa N.
Numerade Educator

Problem 32

Use either a computer algebra system or a table of integrals to find the $ exact $ length of the arc of the curve $ y = x^{\frac{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.

Clarissa N.
Numerade Educator

Problem 33

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

Clarissa N.
Numerade Educator

Problem 34

(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) $.

Clarissa N.
Numerade Educator

Problem 35

Find the arc length function for the curve $ y = 2x^{\frac{3}{2}} $ with starting point $ P_0 (1, 2) $.

Clarissa N.
Numerade Educator

Problem 36

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

Clarissa N.
Numerade Educator

Problem 37

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

Clarissa N.
Numerade Educator

Problem 38

The arc length function for a curve $ y = f(x) $, where $ f $ is an increasing function, is $ s(x) = \int_0^x \sqrt{3t + 5}\ dt $.
(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.

Clarissa N.
Numerade Educator

Problem 39

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 N.
Numerade Educator

Problem 40

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.

Clarissa N.
Numerade Educator

Problem 41

A hawk flying at 15 m/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.

Clarissa N.
Numerade Educator

Problem 42

The Gateway Arch in St. Louis (see the photo on page 543) was constructed using the equation
$$ y = 211.49 - 20.96 \cosh 0.03291765x $$
for the central curve of the arch, where $ x $ and $ y $ are measured in meters and $ \mid x \mid \le 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 N.
Numerade Educator

Problem 43

A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. high 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 (\frac{\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 N.
Numerade Educator

Problem 44

(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 (\frac{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 N.
Numerade Educator

Problem 45

Find the length of the curve
$ y = \displaystyle \int_1^x \sqrt{t^3 - 1}\ dt $ $ 1 \le x \le 4 $

Clarissa N.
Numerade Educator

Problem 46

The curves with equations $ x^n + y^n = 1 $, $ n = 4, 6, 8 \cdots , $ 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_{2k} $ of the fat circle with $ n = 2k $. Without attempting to evaluate this integral, state the value of $ \lim_{k\to\infty} L_{2k} $.

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