1. Chapter 8 – Further Applications of Integration
8.1 Arc Length
8.1 Arc Length
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2. Further Applications of Integration
In chapter 6, we looked at some applications of integrals:
Areas
Volumes
Work
Average values
8.1 Arc Length
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3. Further Applications of Integration
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 Length
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4. Further Applications of Integration
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 Length
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5. Arc Length
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 Length
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6. Arc Length 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
8.1 Arc Length
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7. Arc Length 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.
8.1 Arc Length
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8. Arc Length
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
Length 8
3.8125 16
3.8183 32
3.8197 64
3.8201
128
3.8202
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9. Length of Curves
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):
8.1 Arc Length
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10. The Arc Length Formula
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
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11. Example 1 – pg. 543 # 12 Find the exact length of the curve.
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12. Arc Length Formula
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
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13. Example 2 – pg. 543 # 10 Find the exact length of the curve.
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14. The Arc Length Function
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
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15. The Arc Length Function
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
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16. Differentials The differential of the arc length then is:
And is sometimes written in the symmetric form:
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17. Example 3 – pg. 543 # 20
Find the length of the arc of the curve from point P to point Q.
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18. Arc Length and Simpson’s Rule
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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19. Example 4 – pg. 543 # 24
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.
8.1 Arc Length
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20. Book Resources Video Examples More Videos Wolfram Demonstrations
Example 2 – pg. 540
Example 3 – pg. 541
Example 4 – pg. 542
More Videos
Arc Length Parameter
Wolfram Demonstrations
Arc Length
7.7 Approximation Integration
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21. Web Resources http://calculusapplets.com/arclength.html
6.html
3.html
1.html
8.1 Arc Length
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