Chapter 8 – Further Applications of Integration 8.1 Arc Length Erickson 8.1 Arc Length 1.
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Transcript of Chapter 8 – Further Applications of Integration 8.1 Arc Length Erickson 8.1 Arc Length 1.
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Chapter 8 – Further Applications of Integration
8.1 Arc Length
Erickson8.1 Arc Length
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8.1 Arc Length2
Further Applications of Integration
Erickson
In chapter 6, we looked at some applications of integrals:
Areas Volumes Work Average values
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8.1 Arc Length3
Further Applications of Integration
Erickson
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
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8.1 Arc Length4
Further Applications of Integration
Erickson
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
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8.1 Arc Length5
Arc Length
Erickson
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.
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8.1 Arc Length6
Arc Length
Erickson
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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8.1 Arc Length7
Arc Length
Erickson
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
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8.1 Arc Length8
Arc Length
Erickson
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
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8.1 Arc Length9
Length of Curves
Erickson
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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8.1 Arc Length10
The Arc Length Formula
Erickson
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
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8.1 Arc Length11
Example 1 – pg. 543 # 12
Erickson
Find the exact length of the curve.
ln cos , 03
y x x
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8.1 Arc Length12
Arc Length Formula
Erickson
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
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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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8.1 Arc Length14
The Arc Length Function
Erickson
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
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8.1 Arc Length15
The Arc Length Function
Erickson
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
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8.1 Arc Length16
Differentials
Erickson
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
Erickson
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
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8.1 Arc Length18
Arc Length and Simpson’s Rule
Erickson
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
Erickson
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
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7.7 Approximation Integration20
Book Resources
Erickson
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
Erickson
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