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Unformatted text preview: 2.5. INTEGRATION ALONG CURVES 197 polygons give a lower bound follows from the much simpler assumption, which we take as selfevident, that the shortest curve between two points is a straight line segment. Suppose that C is an arc parametrized by p : R R 3 and let P = { a = t < t 1 < < t n = b } be a partition of the domain of p . The sum ( P , p ) = n summationdisplay j =1 bardbl p ( t j ) p ( t j 1 ) bardbl is the sum of the straightline distances between successive points along C . It is clear that any reasonable notion of the length of C at least equals ( P , p ). We would also think intuitively that a partition with small mesh size should give a good approximation to the true length of C . We therefore say C is rectifiable if the values of ( P , p ) among all partitions are bounded, and then we define the arclength of C to be s ( C ) = sup P ( P , p ) ....
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 Fall '08
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 Calculus, Polygons

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