Engineering Calculus Notes 209

# Engineering Calculus Notes 209 - 2.5 INTEGRATION ALONG...

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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 self-evident, 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 ℓ ( P , −→ p ) = n summationdisplay j =1 bardbl −→ p ( t j ) − −→ p ( t j − 1 ) bardbl is the sum of the straight-line distances between successive points along C . It is clear that any reasonable notion of the “length” of C at least equals ℓ ( 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 ℓ ( P , −→ p ) among all partitions are bounded, and then we define the arclength of C to be s ( C ) = sup P ℓ ( P , −→ p ) ....
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