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Arc length with vector functions
- Intro Lesson: a6:16
- Intro Lesson: b13:10
- Lesson: 17:42
- Lesson: 23:57
- Lesson: 37:09
- Lesson: 49:07
- Lesson: 59:35
Arc length with vector functions
Lessons
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≤t≤b by calculating:
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≤t≤b by calculating:
L=∫ab∣∣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)=∫0t∣∣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.- IntroductionArc 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)→ 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)=<2+3t,t2,343t23>0≤t≤1
- 2.Determine the length of the vector function on the given interval:
r(t)=(3+4t)i+(2t−3)j+(5−t)k2≤t≤3
- 3.Finding the Arc Length Function
Determine the arc length function for the given vector functionr(t)=<2t,31t3,t2>
- 4.Determine the arc length function for the given vector function
r(t)=<t2,2t2,31t3>
- 5.Finding a Specific Point on a Curve
After traveling a distance of 2π, determine where we are on the vector function r(t)=<cost,sint,t>.