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Unformatted text preview: LESSON 6 Arc Length, Speed, and Curvature READ: Sections 14.3, 14.4, 14.5 Skim section 14.6 NOTES: In section 12.2 of the text, the formula for the length, L, of the plane curve C given by parametric equations ( x = x ( t ) y = y ( t ) t [ a,b ] was found to be L = Z b a p x ( t ) 2 + y ( t ) 2 dt. Mimicking the reasoning that produced that formula leads to an arc length formula for curves in space. In fact, lets make the following definition: if a curve C in 3-space is given parametrically by x = x ( t ) y = y ( t ) t [ a,b ] z = z ( t ) the length of C is given by the formula L = Z b a p x ( t ) 2 + y ( t ) 2 + z ( t ) 2 dt. If the vector form of the equation of C is used,- r ( t ) = x ( t )- + y ( t )- + z ( t )- k ,t [ a,b ], then noting that the square root in the integrand of the formula is exactly the length of the vector- r ( t ), the formula can be written in the compact form L = Z b a ||- r ( t ) || dt. Many integrals for arc length are quite formidable, and in real life you should expect to use Simpsons Rule, or some such approximating method, to estimate the length of a curve. Most of the problems in the text have been selected to make the integrals possible to do exactly, but even then it sometimes requires a little clever algebra to discover an antiderivative. A special function associated with a piecewise-smooth curve, C , is its arc-length function, s ( t ). If C :- r ( t ) = x ( t )- + y ( t )- + z ( t )- k ,t [ a,b ], then, for any t [ a,b ], s ( t ) = Z t a ||- r ( u ) || du. In plain English, s ( t ) is the length of the part of the curve C traced out between times a and t . In other words, s ( t ) keeps a running total of how much arc length has been traced out up to time t . 1 By differentiating both sides of the equation above that defines s ( t ) (and recalling how to differentiate the integral using the Fundamental Theorem of Calculus) we get the frequently used relation ds dt = ||- r ( t ) || which is often written in the form ds = ||- r ( t ) || dt . In the last lesson, it was pointed out that- T ( t ) =- r ( t ) ||- r ( t ) || gives a unit vector in the direction of the line tangent to the curve C whose vector equation is given by- r . At times it is handy to have a vector that is perpendicular to C at a point on the curve; that is to say, a vector that is perpendicular to the tangent vector at that point. Such a vector is said to be a normal to the curve. If you think about it for a second, you can see that there will always be infinitely many different directions possible for a normal vector; in fact any vector in the plane to which the tangent vector is perpendicular will suffice. Of all of these, one particular normal vector is singled out and called the normal vector. Here is how it is produced: Since- T ( t ) is a unit vector,- T ( t ) - T ( t ) = 1. Differentiating both sides, we see 2) = 1....
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