1 / 14

Further Applications of Integration

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 )

donny
Télécharger la présentation

Further Applications of Integration

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Further Applications • of Integration • 8

  2. 8.1 • Arc Length

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

  4. Arc Length • and since f (xi) –f (xi –1) = f(xi*)(xi –xi –1) • or yi = f(xi*) x

  5. Arc Length • Conclusion: • Which is equal to:

  6. Arc Length formula: • If we use Leibniz notation for derivatives, we can write the arc length formula as follows:

  7. Example 1 • Find the length of the arc of the semicubical parabola • y2 = x3 between the points (1, 1) and (4, 8).

  8. Example 1 – Solution • For 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.

  9. Example 1 – Solution • cont’d • Therefore

  10. 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:

  11. Generalization: • The Arc Length Function

  12. 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:

  13. Example 4 • Find the arc length function for the curve y = x2 – ln xtaking P0(1, 1) as the starting point. • Solution: • If f(x) = x2 – ln x, then • f (x) = 2x –

  14. Example 4 – Solution • cont’d • 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

More Related