In this sectionwill considerthe problemof findingthe lengthof a planecurve.Formulasfor findingthe arcs
of circlesappearedin earlyhistoricalrecordsand they wereknownto manycivilizations.However, very little
was knownaboutfindingthe lengthsof generalcurves,such as the lengthof the curve in the interval
until the discoveryof calculusin the seventeenthcentury.
In calculus,we defineanarc lengthas the lengthof a planecurve over an interval (Figure
17). Whenthe curve has a continuousfirst derivative on we say that is a smoothfunction
(or smoothcurve)on
Figure 17
The Arc LengthProblem
If is a smoothcurveon the interval then the arc length of this curveis definedasExample1:
Find the arc lengthof the curve on (Figure18).