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!
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!!
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!!
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.
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) )?
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.
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.
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)
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.
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
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
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
Wouldn't dl always go from 0 to L, ,from adding nothing to adding the entire arc's length(denoted by L), regardless of whether y is a function of x or vice versa, since that is how dl varies, not it's equivalent expression
Might have to define it parametrically & use L=integral(a to b) sqrt([dx/dt]^2+[dy/dt]^2) dt I honestly don't know though, I'm but a humble meme merchant
@@davidshechtman4746 For a unit circle it is probably the easiest example. The parametricization (parameterization?) you want to take that will go along the same path as the unit circle (or x^2 +y^2 = 1) is for x(t) = cos(t) and y(t) = sin(t) for t = 0 to t = 2pi. Since dx/dt = -sin(t), dy/dt = cos(t), you'll have the integral of root( (dx/dt)^2 + (dy/dt)^2) dt from t = 0 to 2pi. Which will give int(root( (-sint)^2 + (cos(t))^2 ) dt, or int (root (sin^2(t) + cos^2(t)) dt, but since cos^2(t) + sin(^2(t) = 1, you'd have int ( dt) for t = 0 to 2pi. The integral for dt is t + c (we can omit the c because this is a definite integral). Evaluated from 0 to 2pi gives 2pi - 0 = 2pi. Like I mentioned this is the simplest case, but for x^2 + xy + y^2 = 1 it is a bit tougher since the parameterization is not straight forward. The curve is some sort of rotated ellipse.
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.
Can you show that the length of the function f(x) = x ^ n as n increases without bound from (0,0) to (1,1) is equal to 2? It is visually obvious, but I could never figure out the integral to apply the limit.
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
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)
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...
So when they are coming up with proofs is a part of that manipulating it to have it in that integral and dx operator at the end format ? If you didn’t have that dx at the end by factoring it out the integral would not work?
i want to ask wat was the meaning of factoring out the dx squared and extract it from root and write it near ? is it necessary? please answer to my question
Curve line . we don't know it's length but we know about interval length and y intersect with curve so we know area and breadth taken as y axis between interval and interval length so use area divided by it so we got length of the curve ????
whilst watching your pi function video you say that for n factorial you need to apply ) l'hospitals rules n times, what about non integer values of n? can you explain or do a video on what exactly applying l'hospitals rule e.g 1/2 times would entail?
The y'(x) is equal to the tangent of the angle of the tangent line, right? So y'(x) = tan(u) where u is the angle, so using this sub and trig identity L = int( sec(u) dx). I know the x and u-worlds are messed in the last one and I didn't simplify it to know where it goes... Is there some geometric reason for L and sec(u) being related to each other or it's just a coincidence?
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)
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!!
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!!
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
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
It's amazing that you explained in 6 minutes what my calculus teacher couldn't clearly explain in 1 hour.
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👍
That intro is perfect
Thanks.
So incredibly clear! Thank you so much for creating these fantastic videos ❤
Glad you like them!
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.
Bro your video is so funny I kept smiling watching it - while learning a lot! Thanks!
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.
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.
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!
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.
I absolutely love your videos man. You are the best math TH-camr I know and recommend you to anyone I can.
I was searching for a video like this some weeks ago, so happy you uploaded it, thank you
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
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.
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.
Wow, you are doing a great a job by making us understand complex topics like these.🙂
Amazing teachers like you make me love maths even more , thank you
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)
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.
Lowkey flexing with the supreme 👀👀
Now, I can solve any problem regrading this. You made the basics. Thank you.
This is the simplest way I've seen it explained!
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.
3:54 I’m really confused about him factoring (dx)^2.
You sir, deserve a medal. Great explanation 👍👌
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
Good explanation and straight to the point. Thank you for the video!
God I love your enthusiasm
could you make a video deriving the arc length for polar curves too?
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.
I would like to notice we can have the same formula if we have x=h(t) and y=h'(t)
dy/dx = (dy/dt) * (dx/dt)
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
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
Can you found the equal area circle ?
Radius is what so we found percentage of curve length between interval ?
Could be fun with some arc battles.
Also thank you for your videos.
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.
great explanation, but could you follow up with a practical example, for instance, computing the arclength of sin(x) between 0 and pi?
You're awesome! I appreciate your enthusiasm!
loves the explanation, short and clear
Wouldn't dl always go from 0 to L, ,from adding nothing to adding the entire arc's length(denoted by L), regardless of whether y is a function of x or vice versa, since that is how dl varies, not it's equivalent expression
Please derive surface area of cone, cylinder, sphere using surface integral around axis of rotation
Pythagoras theorem and integral calculus - both are required to find arc length. Isn't amazing?
What if its an implicit function, x isn't a function of y, and y isn't a function of x?
Like x^2 + xy + y^2 = 1
Might have to define it parametrically & use L=integral(a to b) sqrt([dx/dt]^2+[dy/dt]^2) dt
I honestly don't know though, I'm but a humble meme merchant
Yep, parametric equations by letting x and y be functions of t.
@@abstractcafe2352 Cool. I don't suppose I could humbly ask you to demonstrate this for a unit circle? Thank you for your time and attention. Dave
@@davidshechtman4746 no point, you can just use the circumference formula & it comes out to be 2pi
@@davidshechtman4746 For a unit circle it is probably the easiest example. The parametricization (parameterization?) you want to take that will go along the same path as the unit circle (or x^2 +y^2 = 1) is for x(t) = cos(t) and y(t) = sin(t) for t = 0 to t = 2pi. Since dx/dt = -sin(t), dy/dt = cos(t), you'll have the integral of root( (dx/dt)^2 + (dy/dt)^2) dt from t = 0 to 2pi. Which will give int(root( (-sint)^2 + (cos(t))^2 ) dt, or int (root (sin^2(t) + cos^2(t)) dt, but since cos^2(t) + sin(^2(t) = 1, you'd have int ( dt) for t = 0 to 2pi. The integral for dt is t + c (we can omit the c because this is a definite integral). Evaluated from 0 to 2pi gives 2pi - 0 = 2pi.
Like I mentioned this is the simplest case, but for x^2 + xy + y^2 = 1 it is a bit tougher since the parameterization is not straight forward. The curve is some sort of rotated ellipse.
2:47 , the limits would be { 0 to L }
this can be applied to parametric equations aswell i assume? just doing extra steps to be in "t" (if x = f(t), y = g(t)) ?
Trystan Hooper
Yes. I did that as well. The video will be up soon
isn't L = integ (from 0 to 1) dL ?
or from n to n+1 ?
thank you for making this video .
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.
Can you show that the length of the function f(x) = x ^ n as n increases without bound from (0,0) to (1,1) is equal to 2? It is visually obvious, but I could never figure out the integral to apply the limit.
Nice intro!!
Wow, this is amazing!
You just saved me bro. I love you!
hello, i really thank you for this proof (why can't i see it ?)
is surface element has relation with unit tangent vector
This is easily the simplest way I've seen of deriving the formula.
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
best teacher ever
show time dilation between two points as direct length and curve length of various type like parabola or circle or any other geometric figures
Thank you! My book was not clear in how this formula came about.
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
Thank you so muchhhh😍😭 you‘re much better than my uni lecturer😍
Say "ruler" 10 times in a row
Sir you know the importance of understanding 👍❤️
2:22 “anyway here’s the deal”
2:24 “this is the ‘d L’ “
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.
there are more cases fe
curve given by parametric equations
curve in polar coordinates
Love the Doraemon theme in the background
Perfect explanation
fantastic explanation
Hi, do you have a video on how to graph a cycloid and an epicycloid given a their parametric equations? thanks a lot !
You explain this perfectly. Thank you!
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!!
please could you do a vid about the area of a 3D curve? that should be very interesting
I keep thinking he’s saying “this is the deal..”😂
Wow😲😲 never thought of this
why do you take the positive root of (dx)^2 when you are taking it out of the square root?
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...
So when they are coming up with proofs is a part of that manipulating it to have it in that integral and dx operator at the end format ? If you didn’t have that dx at the end by factoring it out the integral would not work?
It would be awesome if you did an actual problem for both cases?
i want to ask wat was the meaning of factoring out the dx squared and extract it from root and write it near ? is it necessary?
please answer to my question
I would have expected this video to include parameterized curves as well... As you might have for curves where y isn't a function of x nor x of y.
Curve line .
we don't know it's length but we know about interval length and y intersect with curve so we know area and breadth taken as y axis between interval and interval length so use area divided by it so we got length of the curve ????
So how can this be applied to find the arc length on a circle from one x-coordinate to another?
Your videos are addictive
Best teacher
You helped me a lot thank you!
Great explanation.
U'r so simple i liked that soo much❤️❤️❤️
very good explanations
Thank you so much!! you're a hero 💗💗💗💗👍
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.
whilst watching your pi function video you say that for n factorial you need to apply ) l'hospitals rules n times, what about non integer values of n? can you explain or do a video on what exactly applying l'hospitals rule e.g 1/2 times would entail?
The y'(x) is equal to the tangent of the angle of the tangent line, right? So y'(x) = tan(u) where u is the angle, so using this sub and trig identity L = int( sec(u) dx). I know the x and u-worlds are messed in the last one and I didn't simplify it to know where it goes... Is there some geometric reason for L and sec(u) being related to each other or it's just a coincidence?
how we findout arc length and original function relationship coefficient???