How long is a curve?? The Arclength Formula in 3D

What is the arclength of a vector-valued function or curve in 3D? In this video we break the length into a sum of little straight lines, we add up the lengths of the straight lines, take a limit and voila, we get an integral formula to compute arclength.

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This video was created by Dr. Trefor Bazett

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[moliveau]

I've never commented on a youtube video before but I had to share my love for how beautiful you explained this concept. THANK YOU!

[ntruesdale]

This is one of the most amazing ‘moves’ I’ve ever seen in math. You are, ever so cleverly, finding the length of a curve in space, by converting the problem to finding the volume under a surface. This is actually done quite naturally when you realize that the approach of an infinite series is identical to an integral.

I learned this about a year ago with a curve in the plane, and used Desmos to plot both that curve and the curve whose ‘integration area’ was equal to the length of the main subject. It just makes you smile to see them together.

I’m actually not sure about the length of a curve in 3-space being the equivalent of volume under a surface. That’s just my intuition based on the length of a curve in the plane being equal to the area under some other curve. That is definitely a thing (that makes me smile, as stated).

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I've been serching for an explanation about this topic for a quite long time, and so far this is the best and most detailed explanation. Thank you so much.

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Great choice of colours - curves in blue, line segments in red and purple for points on the blue curves really brings clarity to the graph. Wonderfully done!

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nice! i liked how you talked about the derivation and not just the formula - not only that but the derivation was elegantly explained

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