13.3 Arc Length and Unit Tangent Vector T

Definition Length of A smooth Curve

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The length of a smooth curve , that is traced exactly once as t increases from t =a to t =b , is

Arc Length Formula

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Example:1 Find the length of the curve

Arc Length Parameter with base Point

Speed on a Smooth Curve Definition: Unit Tangent VectorThe unit tangent vector of a smooth curve

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is T =

Example:2 Given Find (i) the length of the curve within the given t interval. (ii) the unit tangent vector of the curve .

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