8 Further Applications of Integration

8.1 Arc Length

33

Arc Length

Suppose that a curve C is defined by the equation y = f (x) where f is continuous and a x b.

44

Arc Length

and since f (xi) – f (xi –1) = f (xi* )(xi – xi –1)

or yi = f (xi* ) x

55

Arc LengthConclusion:

Which is equal to:

66

Arc Length formula:

If we use Leibniz notation for derivatives, we can write the arc length formula as follows:

77

Example 1

Find the length of the arc of the semicubical parabola

y2 = x

3 between the points (1, 1) and (4, 8).

88

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.

99

Example 1 – SolutionTherefore

cont’d

1010

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:

1111

Generalization:The Arc Length Function

1212

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:

1313

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

1414

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