Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
Example 2.2.3
Calculate the length of the curve defined parametrically by , , for .
Solution
Mathematical Solution
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 = .
Figure 2.2.3(a) provides a graph of the given curve.
Figure 2.2.3(a) Graph of the given curve
For the given curve,
=
The arc length is then
Maple Solution - Interactive
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.
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Define the helix as the position vector R
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
Write and evaluate the arc-length integral
Calculus palette: Definite integral template
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the helix as the position vector R.
Apply the int, Norm, and diff commands.
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