Arc Length of a Space Curve + Example on Circular Helix

mes (67)in DTube • 2 days ago

In this video, I extend the formula for the arc length of a 2D plane curve to that of a 3D space curve, both of which can be represented in compact form as the integral of the length of the derivative of the vector function that represents the curve. Although I recap on the derivation for the plane curve arc length I did in my previous videos, I do not go over the proof for the space curve length in this video; perhaps in a future video.

I also go over an example on determining the length of a section of a circular helix, as well as plotting it in GeoGebra's amazing 3D graphing calculator: https://www.geogebra.org/3d/gbvmtdug

#math #vectors #calculus #arclength #geogebra

Timestamps:

Arc Length of a plane curve with parametric equations – 0:00
Recap on Arc Length derivation from my earlier videos – 1:43
Arc Length of a space curve is defined in the same way as for plane curves – 5:55
Compact form of the arc length formulas as the integral of the length or magnitude of the derivative of the vector function – 9:28
Recap on the distance formula in 3D (and 2D) – 13:08
Example 1: Arc Length of a Circular Helix –14:11
- Arc length = 2(2^(1/2)pi ≈ 8.89 – 17:40
- Graph of the helix from Calculus Book and from GeoGebra – 19:27

Notes and playlists: