Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
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Example 2.2.5
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Obtain , the arc-length function for the curve in Example 2.1.3.
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Solution
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Mathematical Solution
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If the position-vector description of a curve is given by , then , where the over-dot notation represents differentiation with respect to . Hence, the integrand in the arc-length integral for R is = .
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=
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Hence, the arc-length function is . (Note that because the upper limit of integration is , the integration variable itself must be some other variable, here chosen to be .)
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Maple Solution - Interactive
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Within the Student MultivariateCalculus package, the differentiation operator automatically maps onto the components of vectors. Also, in this package, the norm of a vector defaults to the Euclidean norm.
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Using as the parameter on the helix, define this curve as the position vector R
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Context Panel: Assign to a Name≻R
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Write and evaluate the arc-length integral
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Calculus palette: Definite integral template
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Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Positive
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=
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The arc-length function is therefore . (Note that because the upper limit of integration is , the integration variable itself must be some other variable, here chosen to be .)
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Maple Solution - Coded
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Install the Student MultivariateCalculus package.
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Define the helix as the position vector R.
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Apply the int, Norm, and diff commands.
Note the need for a positivity assumption on .
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=
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The arc-length function is therefore . (Note that because the upper limit of integration is , the integration variable itself must be some other variable, here chosen to be .)
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