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Unformatted text preview: Section 14.3 Arc Length and Curvature Calculus on Curves in Space In this section, we lay the foundations for describing the movement of an object in space. 1. Vector Function Basics In Calc 2, a formula for arc length in terms of parametric equations (in 2-space) was determined. A similar formula holds for 3-space. Result 1.1. If vector r ( t ) = f ( t ) vector i + g ( t ) vector j + h ( t ) vector k defines a smooth curve C and f ( t ), g ( t ) and h ( t ) are all continuous, then the arc length along the portion of the curve with a lessorequalslant t lessorequalslant b (provided it is traversed only is) is given by the formula L = integraldisplay b a radicalbig [ f ( t )] 2 + [ g ( t )] 2 + [ h ( t )] 2 dt or L = integraldisplay b a || vector r ( t ) || dt. The formula is straight forward to work with: Example 1.2. Find the length of vector r ( t ) = vector i + t 2 vector j + t 3 vector k for 0 lessorequalslant t lessorequalslant 1. This is straight forward calculations: L = integraldisplay 1 2 + 4 t 2 + 9 t 4 dt = integraldisplay 1 2 t 4 + 9 t 2 dt. Substituting u = 4 + 9 t 2 , we have du/dt = 18 t , so dt = du/ 18 t , and when t = 0, u = 4, and t = 1 gives u = 13, so integraldisplay 1 2 t 4 + 9 t 2 dt = integraldisplay 13 4 2 t u 1 18 t du = 1 9 integraldisplay 13 4 udu = 1 9 bracketleftbigg 2 3 u 3 2 bracketrightbigg 13 4 = 2 27 bracketleftbigg 13 3 2 4 3 2 bracketrightbigg . Often we are not interested in a specific distance a particle has traveled, but rather a formula to determine how far a particle has traveled in terms of some variable (this is useful in things like airline flight or space travel), so we need to determine a way to do this. Suppose vector r ( u ) is a vector function for a curve C which traverses C only once for a lessorequalslant u lessorequalslant t 1 2 (notice we have changed the parameter to u and we are using t as a time variable). We define the arclength function s for C by s ( t ) = integraldisplay t a || vector r ( u ) || du = integraldisplay t a radicalbig [ f ( u )] 2 + [ g ( u )] 2 + [ h ( u )] 2 du. Notice that the integral defines a function in the variable t . Also note that by the fundamental Theorem of Calculus, this implies ds dt = || vector r ( t ) || . The function s ( t ) measures the distance from the point vector r ( a ) to the point vector r ( t ), so as soon as we plug in a value for t , it will provide us with this numerical distance. We illustrate. Example 1.3. Find the distance formula for vector r ( u ) = 2 u vector i + (1 3 u ) vector j + (5 + 4 u ) vector k from u = 0. Use the formula to determine the distance traveled after 2 seconds, and how long it takes for the particle to travel 8 units....
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