Engineering Calculus Notes 211

Engineering Calculus Notes 211 - 199 2.5. INTEGRATION ALONG...

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Unformatted text preview: 199 2.5. INTEGRATION ALONG CURVES Then b a → − (t) dt − ℓ (P , − ) < 3δ(b − a). → ˙ p p Proof. For the moment, let us fix an atom Ij = [tj −1 , tj ] of P . Applying the Mean Value Theorem to each of the component functions, there exist parameter values si , i = 1, 2, 3 such that x(tj ) − x(tj −1 ) = x(s1 )δj ˙ y (tj ) − y (tj −1 ) = y (s2 )δj ˙ z (tj ) − z (tj −1 ) = z (s3 )δj . ˙ Then the vector → − = (x(s ), y (s ), z (s )) vj ˙1˙2˙3 satisfies → − (t ) − − (t ) = − δ → → pj p j −1 vj j and hence →(t ) − − (t ) = − δ . − → → pj p j −1 vj j But also, for any t ∈ Ij , → − (t) − − ≤ |x(t) − x(s )| + |y (t) − y (s )| + |z (t) − z (s )| → ˙ ˙ ˙1 ˙ ˙2 ˙ ˙3 p vj < 3δ Now, an easy application of the Triangle Inequality says that the lengths of two vectors differ by at most the length of their difference; using the above this gives us → → − ( t) − − ˙ < 3δ for all t ∈ I . v p j 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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