Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
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Example 2.2.6
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a)
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Obtain the arc-length function for the curve , where
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b)
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Invert to obtain and reparametrize the curve with the arc length as the parameter.
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c)
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Show that .
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Solution
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Mathematical Solution
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Part (a)
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Figure 2.2.6(a) provides a graph of the given curve.
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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 = , which becomes
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Figure 2.2.6(a) Graph of the given plane curve
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The arc-length function is then , where the variable of integration is chosen as because the upper limit of integration is itself .
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Part (b)
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To obtain from , solve for by the quadratic formula.
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The result is , but the restriction means that .
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Obtain as follows.
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Part (c)
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The relevant calculations are as follows.
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Maple Solution - Interactive
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Part (a)
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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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Define the 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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Using vertical strokes for norm bars and the Calculus palette for the differentiation operator, write the norm of .
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Constructions≻Definite Integral≻p ≻ Set range from 0 to t
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Context Panel: Evaluate Integral
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Interpret the result as .
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Part (b)
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Write the equation
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Context Panel:
Solve≻Obtain Solutions for≻
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Control-drag the solution with the positive radical.
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Context Panel: Assign to a Name≻
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Re-parametrize R by making the substitution
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Write R, the name of the position vector , and press the Enter key.
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Context Panel: Evaluate at a Point≻
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Context Panel: Simplify≻Simplify
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Context Panel: Assign to a Name≻Rs
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Part (c)
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Calculus palette: Differentiation operator
Apply to
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Simplify
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Maple Solution - Coded
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Part (a)
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Install the Student MultivariateCalculus package.
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Define the curve as the position vector R.
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Apply the int, Norm, and diff commands, imposing the positivity assumption on .
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The result of these calculations is , or .
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Part (b)
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Use the solve command to obtain from the equation .
Assign the sequence of solutions to the name .
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Apply the eval command to obtain .
In addition, apply the simplify command.
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Part (c)
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Apply simplify to the Norm command applied to , obtained in turn by an application of the diff command.
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