Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 72and hence√ (dx ) dr2 ∥dt∥ = +dt( ) 2 dyordt√ (dx ) dr2 ∥dt∥ = +dt( ) 2 dy+dt( ) 2 dzdtSubstituting these expressions in (4.22) and (4.24) leads us to the following theorem.Theorem 4.6 If C is the graph in 2-space or 3-space of a smooth vector-valued functionr(t), then its arc length L from t = a to t = b isL =Example 4.17 Find the arc length of the vector-valued functionfrom t = 0 to t = π.Solution .........∫ badr∥dt∥ dt (4.25)r(t) = costi+sintj+tkExample 4.18 Find the arc length of the vector-valued functionSolution .........Arc Length as a Parameterr(t) = 〈2t,lnt,t 2 〉 for 1 ≤ t ≤ e.For many purpose the best parameter to use for representing a curve in 2-space or 3-spaceparametricallyisthelengthofarcmeasured alongthecurve fromsomefixed reference point.This can be done as follows:Using Arc Length as a ParameterStep 1. Select an arbitrary point on the curve C to serve as a reference point.Step 2. Starting from the reference point, choose one direction along the curve to be thepositive direction and the other to be the negative direction.Step 3. If P is a point on the curve, let s be the signed arc length along C from thereference point to P, where s is positive if P is in the positive direction from thereference point and s is negative if P is in the negative direction.