Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
Example 2.2.2
Calculate the length of the curve defined in Example 2.1.5.
Solution
Mathematical Solution
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 2.2.2(a) provides a graph of the given curve.
For the given curve,
R.
= ddtt2+ddtlnt2+ddtet/32
= 1+1/t2+e2 t/3/9
The arc length is then ∫151+1/t2+e2 t/3/9 ⅆt ≐ 5.9897, where numeric evaluation of the integral is the only recourse.
Figure 2.2.2(a) Graph of the given curve
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. (Be sure to use the exponential "e".)
Context Panel: Assign to a Name≻R
t,lnt,ⅇt/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
Context Panel: Approximate≻5 (digits)
∫15ⅆⅆ t R ⅆt = ∫159+9t2+ⅇt323ⅆt→at 5 digits5.9897
Arc-length integrals are generally the most challenging integrals that arise in a calculus course. For the given curve, the integral cannot be evaluated exactly by Maple, so a numeric evaluation is the appropriate alternative.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the curve as the position vector R. (Be sure to use the exponential "e".)
R≔t,lnt,ⅇt/3:
Apply the evalf, Int, Norm, and diff commands.
evalfIntNormdiffR,t,t=1..5 = 5.989672154
Knowing that Maple's int command will fail to find an exact antiderivative, it is best to use the Int command to form the unevaluated integral, which is then evaluated numerically by the evalf command. If the evalf command is applied to an integral formed by the int command, Maple will first try to find the exact antiderivative before giving up and reverting to numeric integration.
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