Arc length with vector functions

Arc length with vector functions

Lessons

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 atba \leq t \leq b by calculating:

L=abr(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)=0tr(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.
  • Introduction
    Arc Length with Vector Functions Overview:
    a)
    Arc Length
    • Length of a vector function
    • Example of finding the length

    b)
    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))


  • 1.
    Finding the Arc Length
    Determine the length of the vector function on the given interval:

    r(t)=<2+3t,t2,433t32>0t1 r(t) = \lt 2 + 3t, t^2, \frac{4\sqrt{3}}{3} t^{\frac{3}{2}} \gt \;\; 0 \leq t \leq 1


  • 2.
    Determine the length of the vector function on the given interval:

    r(t)=(3+4t)i+(2t3)j+(5t)k2t3r(t) = (3+4t)i + (2t-3)j+(5-t)k \;\; 2 \leq t \leq 3


  • 3.
    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


  • 4.
    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


  • 5.
    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)=<cost,sint,t>r(t)= \lt \cos t, \sin t, t\gt.