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Further Applications of Integration. 8. 8.1. Arc Length. Arc Length. Suppose that a curve C is defined by the equation y = f ( x ) where f is continuous and a x b. Arc Length. and since f ( x i ) – f ( x i – 1 ) = f ( x i * )( x i – x i –1 ) - PowerPoint PPT Presentation
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8 Further Applications of Integration
8.1 Arc Length
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Arc Length
Suppose that a curve C is defined by the equation y = f (x) where f is continuous and a x b.
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Arc Length
and since f (xi) – f (xi –1) = f (xi* )(xi – xi –1)
or yi = f (xi* ) x
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Arc LengthConclusion:
Which is equal to:
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Arc Length formula:
If we use Leibniz notation for derivatives, we can write the arc length formula as follows:
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Example 1
Find the length of the arc of the semicubical parabola
y2 = x
3 between the points (1, 1) and (4, 8).
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Example 1 – SolutionFor the top half of the curve we have y = x3/2
So the arc length formula gives
If we substitute u = 1 + , then du = dx.When x = 1, u = ; when x = 4, u = 10.
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Example 1 – SolutionTherefore
cont’d
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Arc Length (integration in y)
If a curve has the equation x = g (y), c y d, and g (y) is continuous, then by interchanging the roles of x and y we obtain the following formula for its length:
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Generalization:The Arc Length Function
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The Arc Length Function
Given a smooth y = f (x), on the interval a x b ,the length of the curve s(x) from the initial point P0(a, f (a)) to any point Q (x, f (x)) is a function, called the arc length function:
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Find the arc length function for the curve y = x2 – ln x taking P0(1, 1) as the starting point.
Solution:If f (x) = x2 – ln x, then
f (x) = 2x –
Example 4
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Thus the arc length function is given by
Application to a specific end point: the arc length along the curve from (1, 1) to (3, f (3)) is
cont’dExample 4 – Solution