# Arc length with vector functions

### Arc length with vector functions

#### Lessons

Notes:

Finding the Arc Length
Given a vector function $r(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)= $, 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.
• 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?
• $s(t)\to$ The distance travelled on the curve from 0 to $t$
• Example of calculating $s(t)$ and $r(t(s))$

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

$r(t) = \lt 2 + 3t, t^2, \frac{4\sqrt{3}}{3} t^{\frac{3}{2}} \gt \;\; 0 \leq t \leq 1$

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

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

• 3.
Finding the Arc Length Function
Determine the arc length function for the given vector function

$r(t) = \lt 2t, \frac{1}{3} t^3 , t^2 \gt$

• 4.
Determine the arc length function for the given vector function

$r(t) = \lt t^2, 2t^2, \frac{1}{3} t^3 \gt$

• 5.
Finding a Specific Point on a Curve
After traveling a distance of $\sqrt{2} \pi$, determine where we are on the vector function $r(t)= \lt \cos t, \sin t, t\gt$.