w3-C - Math 20C Multivariable Calculus Lecture 7 1 ' $ The...

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Math 20C Multivariable Calculus Lecture 7 1 Slide 1 $ % The arc length of a curve in space Arc length of a curve. Arc length function. Examples. Slide 2 $ % The arc length of a curve is a number that measures the extension of the curve DeFnition 1 Let r ( t ) be a continuously diferentiable vector-valued Function. The length oF the curve associated with r ( t ) For t [ a, b ] is the number given by ba = Z b a | r 0 ( t ) | dt.
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Math 20C Multivariable Calculus Lecture 7 2 Slide 3 $ % The arc length of a curve in space has the following form in components r ( t ) = h x ( t ) , y ( t ) , z ( t ) i , r 0 ( t ) = h x 0 ( t ) , y 0 ( t ) , z 0 ( t ) i , ba = Z b a q [ x 0 ( t )] 2 + [ y 0 ( t )] 2 + [ z 0 ( t )] 2 dt. Suppose that the curve represents the path traveled by a particle in space. Then, the length of the curve is the integral of the speed, | v ( t ) | . So in this case the length of the curve is the distance traveled by the particle. Slide 4
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w3-C - Math 20C Multivariable Calculus Lecture 7 1 ' $ The...

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