Arclength Formula | Derivation & Ex: Circumference of a Circle

Play along with the animations from the video with this DESMOS link, adjusting the sliders for n,a,b or even the function to see how we approximate the curve: www.desmos.com/calculator/h4r...
Description: We can use calculus to compute the arclength of differentiable curves. In this video we develop the formula from basic ideas of integral calculus. Then, knowing the formula, we apply it in a special case, computing the circumfrence of a circle. Of course we have long since memorized that formula, but isn't it nice to see it actually derived?
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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.
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[@sau002]

Excellent. A visual approach like this makes it much easier.

[@jan-willemreens9010]

Dear Dr. Trefor, Because of a logical and step-by-step way you have explained/derived the Arclength Formula, even someone with less mathematical knowledge can understand this, so to speak. Very well done Dr. Trefor and thank you!

[@jan-willemreens9010]

Dear Dr. Trefor, Thank you very much for your quick reply. I often wonder how someone like you for instance with so much knowledge looks at everyday life; is it still possible to observe life events with a neutral view? Maybe an impertinent question of me, in that case I apologize sincerely! Well-balanced, educational and enjoyable math videos, Dr. Trefor. greetings from the other side of the atlantic sea...

[@vihangasathsara612]

Thank you very much professor for teaching these lessons so clearly. Now I can understand the entire arclength lesson easily than before

[@khalidhossain6738]

Your contents are like paid course but you're giving it free. Lots of love from Bangladesh.

[@michael654]

Really appreciate your videos. It's great to watch a video like this before reading the text or attempting problems.

[@somayahbird4527]

I was sick and could not attend calculus2 for for two weeks and your videos helped me a lot. Thank you sir

[@salmael_badry2926]

I'm grateful for this amazing way of explanation.

[@mohammadfallahzade2110]

I just want to say thank you for your time and great work love ❤️ from Iran

[@mathhack8647]

Amazing, well Presented and explained. Thanks.

[@user-qs1tp1ll9i]

This is really the best I have seen that explained how to calculate arc length so awesome.

[@shifagoyal8221]

With graph and examples , concept is easy to grasp.

[@Sarah-tl8cd]

Great video, very succinct

[@019mj]

I'm so grateful to u!

[@dktchr3332]

Nicely done.

[@madisonmobach1205]

Love you Trefor youre the best 😊

[@shubhamsahu492]

it was awesome sir ,please keep making such wonderfull videos ,we are always with you 😤🤩🤩👍🏻👍🏻♥️♥️❤️

[@joyanbhathena7251]

Thanks for this!!

[@andrewlee8340]

You and professor Leonard are currently saving my calc 3 grade

[@jamesperalta2870]

Great content!

[@tankokping1867]

Thumb up to your video, I think you are indeed a good math educator

[@DrTrefor]

Thank you! 😃

[@kaicaspelherr9876]

a quick way to understand the formula also comes from the idea that, if you move along a curve, distance comes from integrating speed, i.e. magnitude of velocity. Understanding that this implies length comes from integrating the magnitude of changes in x and y can allow you to extrapolate to the formula pretty quickly.

[@soonerbjj]

Great video.

[@wakeawake2950]

Nice explanation 👌👍..

[@premgurung7250]

Excellent

[@zaheeruddin1837]

Thnks for this helpful vedio

[@azmffstatus2808]

sir how f'(x) replaced f'(xi*) ???

[@samedbey3548]

Reimann sum. You can pick any x as long as x is in the interval delta_x. The result is the same.

[@ycombinator765]

Respect++ Earned😍😍😍 ❤️ from Pakistan🇵🇰

[@passager683]

That division by zero almost pokes the eye 😂

[@ramizhossain9082]

Great.

[@socrates4730]

U r awesome .

[@muktabhandari8776]

How do you make such a wonderful videos? Any tips.

[@TheStrafendestroy]

Hello Dr. Bazett, I was going to ask why this formula was different than the one for 3D curves with parametric equations, and I think it just clicked why they are different. Here you are converting the change in 'Y' into terms of x because you integrating the curve between some bounds with a change in 'x', but when we are doing parametric curves we need the terms in the form t because we are integrating over some bounds with a change in 't'.

[@aliciacabreraelias793]

hi! can you please help me ? why does he replace the sigma with the integral sign in minute 7:01?

[@carultch]

He took the limit as delta x goes to zero, and n goes to infinity. Just like Riemann sums become integration by doing the same thing when we are first introduced to integration, this sum becomes integration as well. Taking the limit as our segments get small, and our number of segments gets large.

[@duckymomo7935]

deriving this is difficult modeling with differential calculus is still hard, eg trying to derive the differential equation for a one-dimensional wave/string is hard

[@newtonkumarmondol334]

Make a video about fourier series

[@AjaySharma-yh7zr]

Sir, can you please help me, when we have taken the limits for a full Circle, then arclength should be 2π. But that isn't the case here?

[@AjaySharma-yh7zr]

@@DrTrefor Thanks, I missed that

[@aashsyed1277]

what about arc length of a implicit function?

[@federicopagano6590]

8:09 HOW CAN WE TAKE FROM -1 TO 1 IF THE DERIVATIVE IS NOT DEFINDED IN THE BOUNDARIES, I MEAN IF F IS DEFINED OVER -1:1 INCLUDING BOUNDARIES THEN THE DERIVATIVE IS DEFINED OVER THE OPEN INTERVAL, SO HOW CAN WE INCLUDE THE BOUNDARIES WHEN INTEGRATING

[@alirezaghasrimanesh2431]

oh I should train my forearms today