I have a question: Would it be possible for you to derive (show how on heavens earth) the formula of: Int sqrt (a^2-x^2) dx = x/2(sqrt a^2-y^2) - a^2/2(sin^(-1)(x/a))+c Hope I got it right. Found it in a table for a probkem I have but I am sooo lost in the integrationworld. Would be nice to see different derivations with some simple graphics on the board as well. Thank you sir, for your work, it is appriciated all over the world!
I can't tell you how happy I am to have come across your channel. Nobody has explained this concept as clearly as you have. It is so important to understand what the formula stands for and this is right on the money! Thank you so much!!
It takes 7 seconds to skim the proof from the textbook. It took 7 minutes to understand the proof in this video. Absolutely worth it. Amazing job and thank you!!
Very nice video bro. I remember I did the exact same derivation when I was studying calculus, but then realized this derivation is in fact incomplete, because the pits of (dy) are not necessarily equal in length, but the pits of (dx) are, and I saw text books use the mean value theorem in their derivations to overcome that.
Perfect timing. Self teaching my self line integration and this is a great explanation for part of that crazy formula int(f(x(t), y(t))√((dx/dt)^2 + (Dy/dt)^2) dt
Excellent that you identified how the 'elemental length' is constructed in terms of the coordinate space. Getting this firmly grasped is key to tackling the 'bigger stuff' - circle, ellipse, spirals - then onto 3D with helix et al. Please use this episode as a launching point for a series, working upwards through the understanding/complexity of finding arc lengths 'from first principles'. That is what will make the "Aha!" Light Bulb come on in peoples heads and stay there forever.
Yeah, I wish there were channels that teach math of physics at full depth starting from zero instead of just making use of that knowledge to do random stuff that require the view to already know the subject in order to understand what they're talking about.
Thanks for this. Your explanations are brilliant. There's another case when x and y are parameterised. e.g. if you have the circle defined by x(s) = r.cos(s), y(s) = r.sin(s) and you want the arc length between s = 0 and s = 2π dl^2 = dx^2 + dy^2 dx = dx/ds ds = -r.cos(s) ds dy = dy/ds ds = r.sin(s) ds so dl^2 = r^2 (cos^2(s) + sin^2(s)) ds^2 dl = rds L = r∫[0 to 2π] ds = 2πr Please could you show us how to calculate the arc length of an ellipse? ( x(s) = a.cos(s), y(s) = b.sin(s) )?
Seems like a natural followup would be when the curve L is a function over time t from time a to time b (e.g. F(t) = (sin(t), cos(t)) in the cartesian coordinates to describe a circular path) and looking at the integral over dt.
When i was 15 ( almost One year ago) when i was making theories when i was learning university maths at same time and i discovered a Proof to this formula implicitly ( by a diferencial infinite series), but i though i discovered a new thoery but then i realized that my formula is another Proof to this formula😂😂🤦🤦 i got euforic and then depressed after that
for anyone's confused at 3:54 why (dx)^2 + (dy)^2 = (dx)^2 * ( 1 + (dy)^2/(dx)^2) ) since (dy)^2 = (dy)^2 . (dx)^2 / (dx)^2 (which is = 1) u can basically create a dx out of thin air. Then, obviously, we just need to factor the dx out (dx)^2 + (dy)^2 * (dx)^2 / (dx)^2 = (dx)^2 * ( 1 + (dy)^2/(dx)^2)
I wanna challenge you to find a way to calculate the integral that gave π(at least 2 digits) without calculators!!! And remember, before π discovered, you couldn't use it because...
3.5 years late, but... There's also the case when x = f(t), y = g(t). In that case, you need to use the "substitutions" dx = (dx/dt) dt and dy = (dy/dt) dt (these aren't really substitutions, in a rigorous sense, but the symbolic manipulation works, so...). This gives dL = sqrt(f'(t)^2 + g'(t)^2) dt, where the derivatives are taken with respect to t, and the limits of integration will be from the initial value of t to the final value of t. Now for a question of my own: are my comments helpful, or am I just coming across as a smarmy little know-it-all?
I read ur comment 1 year ago but at that time I was unable to understand what about u discussed but it was looking me something helpful so I took screenshot and move on but today when I faced a similar problem then reminds me something that u said so, then I go back to my year ago screenshot list and after reading ur comment again, I realize that the problem I'm faced today , u have explained already a year ago so, my happiness forced me to come here again and thanking u but this time I'm here just for u ,So thanks a lot .....dear.
It would be cool for you to demonstrate the arc length formula with a practical example, like the arc length of the semi circle (x**2+y**2=r**2) and then resolving to pi.
Integrating sums all the tiny DLs which make up the length Why not area i think because we restricted dx and dy, by Pythagoras's theorem to be a length of a hypotenuse which length because tiny as dx approaches 0
And if you have x(t) and y(t) you do the integral sqrt((dx/dt)^2+(dy/dt)^2)dt from t_a to t_b? Ex: x(t)=e^t * cos(t) and y(t)=e^t * sin(t) from 0 to Pi/2
I think you can separate the problem is other smaller problems, for example the length of the circle is 4 times the quarter or the circle which you know is y=sqrt(1-x^2) for example. For the parabola, you have the expression of f(x)=y if you know a few points so it shouldn't be too much of a problem. For other cases like the circle where you have 2 or more images for a single x or y value, just split the problem in several little curves you can then add up and it shouldn't be too hard after this if you know the formula of the curves
I tried the arc length of sin x but I can’t evaluate the integral . Internet says that it is an elliptic intergal, so now I’m wondering what’s an elliptic integral.
Hi, Blackpenredpen. I like your videos and I learned a lot about calculus in your videos (although I'm 15, and we don't do it in school yet :)) I am interested in limits, so I found this one: lim (n-->inf) 4/n*(sqrt(2/n-1/n^2)+sqrt(4/n-4/n^2)+sqrt(6/n-9/n^2)+sqrt(8/n-16/n^2)....+sqrt(2k/n-k^2/n^2)...). Can you compute it? (You can put it in sigma calculator to see how interesting it is)
Minor picky mistake,
*Please write "dL" instead of "dl".*
Because when we integrate dL we will get L.
While integral of dl is l.
Ok sir
@Tigc channel 2 why
I have a question:
Would it be possible for you to derive (show how on heavens earth) the formula of:
Int sqrt (a^2-x^2) dx = x/2(sqrt a^2-y^2) - a^2/2(sin^(-1)(x/a))+c
Hope I got it right. Found it in a table for a probkem I have but I am sooo lost in the integrationworld. Would be nice to see different derivations with some simple graphics on the board as well.
Thank you sir, for your work, it is appriciated all over the world!
dl also means decilitre :)
What an amateur... Unsubbed >:(
2:07 "And now, here is the dL.."
Ahahaha
Pythagoras is always here to solve our problems...
Budhayana*
Better that than Gougu!
Thank you papa Pythagoras 🎉💐
@@fifiwoof1969 bro it’s the same
HAHAHAHAAHAHAHAHAHAHAHAHAHAHA
That is a nice joke
(Seriously, I laght from this Joke)
I can't tell you how happy I am to have come across your channel. Nobody has explained this concept as clearly as you have. It is so important to understand what the formula stands for and this is right on the money! Thank you so much!!
It takes 7 seconds to skim the proof from the textbook. It took 7 minutes to understand the proof in this video. Absolutely worth it. Amazing job and thank you!!
Very good explanation. I'm in disbelief that some people don't like it.
Pathagorean’s!!! They don’t like anyone!!!
Maybe because there were no questions on the vid?!
But the video is still great tho
You’ve helped me so much with my calculus class, you explain all of these complex subjects so well. Thank you!! I’ve subscribed!
Sure👍
It's amazing that you explained in 6 minutes what my calculus teacher couldn't clearly explain in 1 hour.
Love the intro. It's short and clear!
weerman44 thanks!!!!! It was done by Quahntasy!
I was about to say LITERALLY the same lol
@@MarioPlinplin Lol nice :D
I absolutely love your videos man. You are the best math TH-camr I know and recommend you to anyone I can.
Bro your video is so funny I kept smiling watching it - while learning a lot! Thanks!
So incredibly clear! Thank you so much for creating these fantastic videos ❤
Glad you like them!
Amazing teachers like you make me love maths even more , thank you
I was searching for a video like this some weeks ago, so happy you uploaded it, thank you
Very nice video bro. I remember I did the exact same derivation when I was studying calculus, but then realized this derivation is in fact incomplete, because the pits of (dy) are not necessarily equal in length, but the pits of (dx) are, and I saw text books use the mean value theorem in their derivations to overcome that.
New intro by Quahntasy! He is awesome and creative! Check him out th-cam.com/channels/tlaa8gywhvUdrcdYQf5QQQ.html
Thanks again :)
Very clear and concise video, good work!
That intro is perfect
Thanks.
thank you so much, i saved so much time by understanding in just 5 minutes instead of reading a 5 page long of contents inside my textbook.
That was very clear and concise. The textbook sometimes gets very confusing. Now, I can go back and read the textbook again on this chapter.
Haha I worked out the same formula when I did this for fun once. Showed it to my professor and he showed it to the whole class.
Perfect timing. Self teaching my self line integration and this is a great explanation for part of that crazy formula int(f(x(t), y(t))√((dx/dt)^2 + (Dy/dt)^2) dt
You sir, deserve a medal. Great explanation 👍👌
Wow, you are doing a great a job by making us understand complex topics like these.🙂
Excellent that you identified how the 'elemental length' is constructed in terms of the coordinate space. Getting this firmly grasped is key to tackling the 'bigger stuff' - circle, ellipse, spirals - then onto 3D with helix et al.
Please use this episode as a launching point for a series, working upwards through the understanding/complexity of finding arc lengths 'from first principles'. That is what will make the "Aha!" Light Bulb come on in peoples heads and stay there forever.
Exactly!!
Yeah, I wish there were channels that teach math of physics at full depth starting from zero instead of just making use of that knowledge to do random stuff that require the view to already know the subject in order to understand what they're talking about.
Now, I can solve any problem regrading this. You made the basics. Thank you.
Holy, this guy is brilliant! I've seen him once before but only at a glance. So glad I found this video, you don't need to tell me twice to subscribe.
Good explanation and straight to the point. Thank you for the video!
God I love your enthusiasm
This is the simplest way I've seen it explained!
loves the explanation, short and clear
Lowkey flexing with the supreme 👀👀
Thanks for this. Your explanations are brilliant. There's another case when x and y are parameterised.
e.g. if you have the circle defined by x(s) = r.cos(s), y(s) = r.sin(s) and you want the arc length between s = 0 and s = 2π
dl^2 = dx^2 + dy^2
dx = dx/ds ds = -r.cos(s) ds
dy = dy/ds ds = r.sin(s) ds
so dl^2 = r^2 (cos^2(s) + sin^2(s)) ds^2
dl = rds
L = r∫[0 to 2π] ds = 2πr
Please could you show us how to calculate the arc length of an ellipse? ( x(s) = a.cos(s), y(s) = b.sin(s) )?
To find the complete arc length of an ellipse find the quarter arc length (using all positive values), and then multiply it by 4.
You're awesome! I appreciate your enthusiasm!
Could be fun with some arc battles.
Also thank you for your videos.
You just saved me bro. I love you!
thank you for making this video .
best teacher ever
Thank you so muchhhh😍😭 you‘re much better than my uni lecturer😍
Seems like a natural followup would be when the curve L is a function over time t from time a to time b (e.g. F(t) = (sin(t), cos(t)) in the cartesian coordinates to describe a circular path) and looking at the integral over dt.
high school me derived this formula while being in his dad's card and felt happy about it. lol.
When i was 15 ( almost One year ago) when i was making theories when i was learning university maths at same time and i discovered a Proof to this formula implicitly ( by a diferencial infinite series), but i though i discovered a new thoery but then i realized that my formula is another Proof to this formula😂😂🤦🤦 i got euforic and then depressed after that
Great video, well done! If I were you, I wouldn't use dx and dy at start, but *Δx* and *Δy* as they are not infinitesimal.
well obviously he is assuming they are. just blown up for viewing purposes.
Sir you know the importance of understanding 👍❤️
Your videos are addictive
You explain this perfectly. Thank you!
Best teacher
You helped me a lot thank you!
fantastic explanation
Wow, this is amazing!
Thanks man.
You made the topic so easy that every one can understand it...keep on making such videos...
Thank you! Such a clear explanation! Also, the ball in your hand reminds me of the Ood, an alien species of the sci-fi show dr. Who.
U'r so simple i liked that soo much❤️❤️❤️
Thank you so much!! you're a hero 💗💗💗💗👍
Perfect explanation
Nice intro!!
for anyone's confused at 3:54 why (dx)^2 + (dy)^2 = (dx)^2 * ( 1 + (dy)^2/(dx)^2) )
since (dy)^2 = (dy)^2 . (dx)^2 / (dx)^2 (which is = 1) u can basically create a dx out of thin air. Then, obviously, we just need to factor the dx out
(dx)^2 + (dy)^2 * (dx)^2 / (dx)^2 = (dx)^2 * ( 1 + (dy)^2/(dx)^2)
Thank you! My book was not clear in how this formula came about.
Great explanation.
very good explanations
3:54 I’m really confused about him factoring (dx)^2.
This is easily the simplest way I've seen of deriving the formula.
Thank you sensei
Wow😲😲 never thought of this
You rock man !
Thank you so much..much effective 👍 and very clear
Nice work
Thank you so much. You reminded me of using Pythagoras everywhere 🤣
Gogou theorem
Thanks a lot bro for your help.
After relearnijg little segments of math randomly it seems so simple each time lol, but it is hard to remember how to derive all these in the moment
Here is the "dL" lmao, great video!
I wanna challenge you to find a way to calculate the integral that gave π(at least 2 digits) without calculators!!!
And remember, before π discovered, you couldn't use it because...
Amazingly simple
Really nice formula !
Can you found the equal area circle ?
Radius is what so we found percentage of curve length between interval ?
Medio entiendo el inglés, pero se entiende perfectamente lo que explicas. Gracias.
thank you so much sir ❤❤
3.5 years late, but... There's also the case when x = f(t), y = g(t). In that case, you need to use the "substitutions" dx = (dx/dt) dt and dy = (dy/dt) dt (these aren't really substitutions, in a rigorous sense, but the symbolic manipulation works, so...). This gives dL = sqrt(f'(t)^2 + g'(t)^2) dt, where the derivatives are taken with respect to t, and the limits of integration will be from the initial value of t to the final value of t.
Now for a question of my own: are my comments helpful, or am I just coming across as a smarmy little know-it-all?
I read ur comment 1 year ago but at that time I was unable to understand what about u discussed but it was looking me something helpful so I took screenshot and move on but today when I faced a similar problem then reminds me something that u said so, then I go back to my year ago screenshot list and after reading ur comment again, I realize that the problem I'm faced today , u have explained already a year ago so, my happiness forced me to come here again and thanking u but this time I'm here just for u ,So thanks a lot .....dear.
could you make a video deriving the arc length for polar curves too?
It would be cool for you to demonstrate the arc length formula with a practical example, like the arc length of the semi circle (x**2+y**2=r**2) and then resolving to pi.
Love the Doraemon theme in the background
I keep thinking he’s saying “this is the deal..”😂
2:47 , the limits would be { 0 to L }
Why is the arc length equal to integral of dL? Shouldn’t an integral give area?
Integrating sums all the tiny DLs which make up the length
Why not area i think because we restricted dx and dy, by Pythagoras's theorem to be a length of a hypotenuse which length because tiny as dx approaches 0
isn't L = integ (from 0 to 1) dL ?
or from n to n+1 ?
Say "ruler" 10 times in a row
Can you make this for 3D-curve? This must be correct formula, but its not easy to figure out why result is length rather than area.
dope shirt @blackpenredpen
wooow this was awesome mind blown comrade
And if you have x(t) and y(t) you do the integral sqrt((dx/dt)^2+(dy/dt)^2)dt from t_a to t_b? Ex: x(t)=e^t * cos(t) and y(t)=e^t * sin(t) from 0 to Pi/2
Please derive surface area of cone, cylinder, sphere using surface integral around axis of rotation
2:22 “anyway here’s the deal”
2:24 “this is the ‘d L’ “
Its always pythagoras that shows up everywhere, even when you dont expect it...
Thank you very much. 👍👍🔝🔝
Thanks for the video, but now, what about the ellipse or circle?
I think you can separate the problem is other smaller problems, for example the length of the circle is 4 times the quarter or the circle which you know is y=sqrt(1-x^2) for example. For the parabola, you have the expression of f(x)=y if you know a few points so it shouldn't be too much of a problem. For other cases like the circle where you have 2 or more images for a single x or y value, just split the problem in several little curves you can then add up and it shouldn't be too hard after this if you know the formula of the curves
great explanation, but could you follow up with a practical example, for instance, computing the arclength of sin(x) between 0 and pi?
I tried the arc length of sin x but I can’t evaluate the integral . Internet says that it is an elliptic intergal, so now I’m wondering what’s an elliptic integral.
thank you very much
Very very good
Thank you sir
Hi, Blackpenredpen.
I like your videos and I learned a lot about calculus in your videos (although I'm 15, and we don't do it in school yet :))
I am interested in limits, so I found this one: lim (n-->inf) 4/n*(sqrt(2/n-1/n^2)+sqrt(4/n-4/n^2)+sqrt(6/n-9/n^2)+sqrt(8/n-16/n^2)....+sqrt(2k/n-k^2/n^2)...). Can you compute it? (You can put it in sigma calculator to see how interesting it is)
Excellent work young man!!
Thanks sir .
why do you take the positive root of (dx)^2 when you are taking it out of the square root?
So how can this be applied to find the arc length on a circle from one x-coordinate to another?