Arc length with vector functions

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Arc Length with Vector Functions Overview:
Arc Length
- Length of a vector function
- Example of finding the length
Arc Length Function/Why is it Useful?
- $s(t)\to$ The distance travelled on the curve from 0 to $t$
- Example of calculating $s(t)$ and $r(t(s))$

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Finding the Arc Length
Determine the length of the vector function on the given interval:
$r(t) = \lt 2 + 3t, t^2, \frac{4\sqrt{3}}{3} t^{\frac{3}{2}} \gt \;\; 0 \leq t \leq 1$
Determine the length of the vector function on the given interval:
$r(t) = (3+4t)i + (2t-3)j+(5-t)k \;\; 2 \leq t \leq 3$
Finding the Arc Length Function
Determine the arc length function for the given vector function
$r(t) = \lt 2t, \frac{1}{3} t^3 , t^2 \gt$
Determine the arc length function for the given vector function
$r(t) = \lt t^2, 2t^2, \frac{1}{3} t^3 \gt$
Finding a Specific Point on a Curve
After traveling a distance of $\sqrt{2} \pi$ $2 π$ , determine where we are on the vector function $r(t)= \lt \cos t, \sin t, t\gt$ $r (t) =< cos t, sin t, t >$ .

Notes:

Finding the Arc Length
Given a vector function

r(t)= <f(t),g(t),h(t)>

, we can find the arc length of it on the interval

a \leq t \leq b

by calculating:

$L = \int^b_a ||r'(t)||dt$

Finding the Arc Length Function Again, given the vector function

r(t)= <f(t),g(t),h(t)>

, we can find the arc length function

s(t)

by calculating:

$s(t) = \int^t_0 ||r'(u)||du$

Where

s

is the length or distance travelled on the curve in terms of

t

. We usually want to find this if we are looking for

r(t(s))

, which tells us where a point is located on the curve.