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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 )

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Further Applications of Integration

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  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

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