Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
Calculate the length of the helix defined in Example 2.1.4.
If the position-vector description of a curve is given by Rt=xt i+yt j+zt k, then R.=x. i+y. j+z. k, where the over-dot notation represents differentiation with respect to t. Hence, the integrand in the arc-length integral for R is x.2+y.2+z.2 = R..
Figure 2.2.1(a) provides a graph of the given helix.
For the given helix,
Figure 2.2.1(a) Graph of the given helix
The arc length is then ∫03 π10/3 ⅆt=103 3 π=10π.
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
Define the helix as the position vector R
Enter R as per Table 1.1.1.
Context Panel: Assign to a Name≻R
cost,sint,t/3→assign to a nameR
Write and evaluate the arc-length integral
Calculus palette: Definite integral template
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
∫03 πⅆⅆ t R ⅆt = π⁢10
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the helix as the position vector R.
Apply the int, Norm, and diff commands.
intNormdiffR,t,t=0..3 π = π⁢10
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