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!
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...
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
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'.
What is most remarkable is that a text can make up a function where you take the derivative, square it, add one, take the square root, and end up with something you can actually find an anti-derivaitve of in closed form. That is why all the examples in any text look so weird. What about x^3 - 3x or just sine
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.
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.
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
As your n (amount of segments) gets larger, the lenghts of the segments themselves get smaller and smaller. Which means that your interval that you use the mean value theorem for also grows smaller, since its bounded by the endpoints of the line segment. In the limit as n goes to infinity the slope of this line segment approches the slope of the derivative since the point the derivative is at is bounded by the endpoints of the segment AND the segment itself is getting infinitely small. It kind of becomes the same thing. EDIT: idk if you need this anymore or if this is gonna help anyone, but i hope it does lol
Excellent. A visual approach like this makes it much easier.
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!
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...
Your contents are like paid course but you're giving it free. Lots of love from Bangladesh.
Thank you very much professor for teaching these lessons so clearly. Now I can understand the entire arclength lesson easily than before
Really appreciate your videos. It's great to watch a video like this before reading the text or attempting problems.
This is really the best I have seen that explained how to calculate arc length so awesome.
With graph and examples , concept is easy to grasp.
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
I'm grateful for this amazing way of explanation.
Amazing, well Presented and explained. Thanks.
beautiful thankyou what a gorgeous way to proof it loved it😍😍😍😍😍😍
You and professor Leonard are currently saving my calc 3 grade
Thumb up to your video, I think you are indeed a good math educator
Thank you! 😃
Respect++ Earned😍😍😍
❤️ from Pakistan🇵🇰
I was sick and could not attend calculus2 for for two weeks and your videos helped me a lot. Thank you sir
Great video, very succinct
Thanks for this!!
I'm so grateful to u!
it was awesome sir ,please keep making such wonderfull videos ,we are always with you 😤🤩🤩👍🏻👍🏻♥️♥️❤️
That division by zero almost pokes the eye 😂
Nicely done.
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'.
What is most remarkable is that a text can make up a function where you take the derivative, square it, add one, take the square root, and end up with something you can actually find an anti-derivaitve of in closed form. That is why all the examples in any text look so weird. What about x^3 - 3x or just sine
5:29 MVT also requires continuity right
Love you Trefor youre the best 😊
Excellent
I just want to say thank you for your time and great work
love ❤️ from Iran
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.
thanks so much, that's honestly a way better and more intuitive way to understand it.
hi! can you please help me ? why does he replace the sigma with the integral sign in minute 7:01?
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.
Great video.
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
Nice explanation 👌👍..
Thnks for this helpful vedio
Great content!
Make a video about fourier series
How do you make such a wonderful videos? Any tips.
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?
@@DrTrefor Thanks, I missed that
what about arc length of a implicit function?
U r awesome .
sir how f'(x) replaced f'(xi*) ???
Reimann sum. You can pick any x as long as x is in the interval delta_x. The result is the same.
As your n (amount of segments) gets larger, the lenghts of the segments themselves get smaller and smaller. Which means that your interval that you use the mean value theorem for also grows smaller, since its bounded by the endpoints of the line segment. In the limit as n goes to infinity the slope of this line segment approches the slope of the derivative since the point the derivative is at is bounded by the endpoints of the segment AND the segment itself is getting infinitely small. It kind of becomes the same thing. EDIT: idk if you need this anymore or if this is gonna help anyone, but i hope it does lol
Great.
oh I should train my forearms today