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Arc length with vector functions

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Intros
Lessons
  1. Arc Length with Vector Functions Overview:
  2. Arc Length
    • Length of a vector function
    • Example of finding the length
  3. Arc Length Function/Why is it Useful?
    • s(t)→s(t)\to The distance travelled on the curve from 0 to tt
    • Example of calculating s(t)s(t) and r(t(s))r(t(s))
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Examples
Lessons
  1. Finding the Arc Length
    Determine the length of the vector function on the given interval:

    r(t)=<2+3t,t2,433t32>    0≤t≤1 r(t) = \lt 2 + 3t, t^2, \frac{4\sqrt{3}}{3} t^{\frac{3}{2}} \gt \;\; 0 \leq t \leq 1

    1. Determine the length of the vector function on the given interval:

      r(t)=(3+4t)i+(2t−3)j+(5−t)k    2≤t≤3r(t) = (3+4t)i + (2t-3)j+(5-t)k \;\; 2 \leq t \leq 3

      1. Finding the Arc Length Function
        Determine the arc length function for the given vector function

        r(t)=<2t,13t3,t2> r(t) = \lt 2t, \frac{1}{3} t^3 , t^2 \gt

        1. Determine the arc length function for the given vector function

          r(t)=<t2,2t2,13t3>r(t) = \lt t^2, 2t^2, \frac{1}{3} t^3 \gt

          1. Finding a Specific Point on a Curve
            After traveling a distance of 2π\sqrt{2} \pi , determine where we are on the vector function r(t)=<cos⁡t,sin⁡t,t>r(t)= \lt \cos t, \sin t, t\gt.
            Topic Notes
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            Notes:

            Finding the Arc Length
            Given a vector function r(t)=<f(t),g(t),h(t)> r(t)= <f(t),g(t),h(t)>, we can find the arc length of it on the interval a≤t≤ba \leq t \leq b by calculating:

            L=∫ab∣∣r′(t)∣∣dtL = \int^b_a ||r'(t)||dt

            Finding the Arc Length Function Again, given the vector function r(t)=<f(t),g(t),h(t)>r(t)= <f(t),g(t),h(t)>, we can find the arc length function s(t)s(t) by calculating:

            s(t)=∫0t∣∣r′(u)∣∣dus(t) = \int^t_0 ||r'(u)||du

            Where ss is the length or distance travelled on the curve in terms of tt. We usually want to find this if we are looking for r(t(s))r(t(s)), which tells us where a point is located on the curve.