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Chapter 8 – Further Applications of Integration

8.1 Arc Length

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8.1 Arc Length2

Further Applications of Integration

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In chapter 6, we looked at some applications of integrals:

Areas Volumes Work Average values

8.1 Arc Length3

Further Applications of Integration

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Today, we will explore:

Some of the many other geometric applications of integration—such as the length of a curve and the area of a surface

Quantities of interest in physics, engineering, biology, economics, and statistics

8.1 Arc Length4

Further Applications of Integration

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In the next few classes, we will investigate:

Center of gravity of a plate Force exerted by water pressure on a dam Flow of blood from the human heart Average time spent on hold during a customer

support telephone call

8.1 Arc Length5

Arc Length

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Consider the sine curve on the interval [0, π] as shown here.

How can we find the length of this curve? This length is called arc length.

If the curve was a piece of string, we could straighten out the string and then measure the length with a ruler.

8.1 Arc Length6

Arc Length

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But, we don’t have a string to measure! Let’s use line segments because we know how to find the

length of a line segment.

Just a reminder, the length of a line segment from (x1, x2) is

2 2

2 1 2 1x x y y

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Arc Length

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Let’s start by using n = 4. The endpoints of our line

segments are

If we calculate the lengths of each line segment and then add the lengths we will get an approximation for arc length.

We can see from our figure that our estimate is too small. We can improve the estimate by increasing n.

2 3 20,0 , , , ,1 , , , ,0

4 2 2 4 2

3.79L

8.1 Arc Length8

Arc Length

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The table below shows the estimates of the arc length using n line segments. As you would expect, our approximation gets closer to the actual arc length of the curve as n gets larger.

If we let n get arbitrarily large then we could find the actual arc length.

n

n Length

8 3.8125

16 3.8183

32 3.8197

64 3.8201

128 3.8202

8.1 Arc Length9

Length of Curves

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Therefore, we define the length L of the curve C with equation y = f(x), a ≤ x ≤ b, as the limit of the lengths of these line segments (if the limit exists):

11

lim | |n

i in

i

L P P

2 2

1 1 1where i i i i i iP P x x y y

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The Arc Length Formula

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If f ’ is continuous on [a, b], then the length of the curve y = f (x), a ≤ x ≤ b, is

In Leibniz notation we have

21 '( )

b

a

L f x dx

2

1b

a

dyL dx

dx

8.1 Arc Length11

Example 1 – pg. 543 # 12

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Find the exact length of the curve.

ln cos , 03

y x x

8.1 Arc Length12

Arc Length Formula

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If g’ is continuous on [c, d], then the length of the curve x = g(y), c ≤ y ≤ d, is

In Leibniz notation we have

21 '( )

d

c

L g y dy

2

1d

c

dxL dy

dy

8.1 Arc Length13

Example 2 – pg. 543 # 10

Erickson

Find the exact length of the curve.

4

2

1

8 4

yx y

y

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The Arc Length Function

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We will find it useful to have a function that measures the arc length of a curve C from a starting point P(a, f(a)) to any other point on the curve Q(x, f(x)).

If s(x) is the distance along C from P to Q, then, s is a function, called the arc length function, and

2( ) 1 '( )

x

as x f t dt

8.1 Arc Length15

The Arc Length Function

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We can use Part 1 of the Fundamental Theorem of Calculus (FTC 1) to differentiate Equation 5 (as the integrand is continuous):

Which shows that the rate of change of s with respect to x is always at least 1 and is equal to 1 when f ’=0

2

21 '( ) 1

ds dyf x

dx dx

8.1 Arc Length16

Differentials

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The differential of the arc length then is:

And is sometimes written in the symmetric form:

2

1dy

ds dxdx

2 2 2

ds dx dy

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Example 3 – pg. 543 # 20

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

32 4 , (1,5), (8,8)x y P Q

8.1 Arc Length18

Arc Length and Simpson’s Rule

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Because of the square root in the Arc Length formula, sometimes it is very difficult or impossible to evaluate the integral.

In those cases we will try to find an approximation of the length of the curve by using Simpson’s Rule.

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Example 4 – pg. 543 # 24

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Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answers with the value of the integral produced by your calculator.

3 , 1 6y x x

7.7 Approximation Integration20

Book Resources

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Video Examples Example 2 – pg. 540 Example 3 – pg. 541 Example 4 – pg. 542

More Videos Arc Length Parameter

Wolfram Demonstrations Arc Length

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Web Resources

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http://calculusapplets.com/arclength.html

http://youtu.be/PwmCZAWeRNE

http://archives.math.utk.edu/visual.calculus/5/arclength.1/6.html

http://archives.math.utk.edu/visual.calculus/5/arclength.1/3.html

http://archives.math.utk.edu/visual.calculus/5/arclength.1/1.html