Engineering Calculus Notes 218

Engineering Calculus Notes 218 - P = { a = t < t 1 <...

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206 CHAPTER 2. CURVES Integrating a Function along a Curve (Path Integrals) Suppose we have a wire which is shaped like an arc, but has variable thickness, and hence variable density. If we know the density at each point along the arc, how do we Fnd the total mass? If the arc happens to be an interval along the x -axis, then we simply deFne a function f ( x ) whose value at each point is the density, and integrate. We would like to carry out a similar process along an arc or, more generally, along a curve. Our abstract setup is this: we have an arc, C , parametrized by the (continuous, one-to-one) vector-valued function −→ p ( t ), a t b , and we have a (real-valued) function which assigns to each point −→ p of C a number f ( −→ p ); we want to integrate f along C . The process is a natural combination of the Riemann integral with the arclength calculation of § 2.5 . Just as for arclength, we begin by partitioning C via a partition of the domain [ a,b ] of our parametrization
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Unformatted text preview: P = { a = t < t 1 < ··· < t n = b } . ±or a small mesh size, the arclength of C between successive points −→ p ( t j ) is well approximated by △ s j = b −→ p ( t j − 1 ) − −→ p ( t j ) b and we can form lower and upper sums L ( P ,f ) = n s j =1 inf t ∈ I j f ( −→ p ( t )) △ s j U ( P ,f ) = n s j =1 sup t ∈ I j f ( −→ p ( t )) △ s j . As in the usual theory of the Riemann integral, we have for any partition P that L ( P ,f ) ≤ U ( P ,f ); it is less clear that reFning a partition lowers U ( P ,f ) (although it clearly does increase L ( P ,f )), since the quantity ℓ ( P , −→ p ) increases under reFnement. However, if the arc is rectiFable, we can modify the upper sum by using s ( −→ p ( I j )) in place of △ s j ; denoting this by U ∗ ( P ,f ) = n s j =1 sup t ∈ I j f ( −→ p ( t )) s ( −→ p ( I j ))...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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