
Korean: 
우리는 넓이를 계산하기 위해 정적분을 이용했습니다
이제 이 방식을 호의 길이를 구하는 것에
이용하는 방법을 함께 알아보겠습니다
이 것이 무슨 의미일까요?
함수를 그린 그래프에서 
지금 점을 찍은 위치에서 시작을 하고
위쪽 이 점까지 가겠습니다
이 선은 직선이 아닙니다
우리는 이미 직선의 길이를
어떻게 계산할지 알 고 있습니다
대신, 우리는 이 호의 길이를 알고 싶습니다
이 선을 따른 호의 길이를 구한다면
과연 그 길이는 어떻게 될까요?
이것이 제가 이야기 하는 호의 길이 입니다
이는 그래프를 따라 x가 a에서 b까지의 곡선의 길이와
동일하다고 생각하여도 됩니다
이를 어떻게 계산할까요?
적분에서 배운것을 활용하면
이렇게 변화하는 것을 볼 때면
이를 무한히 작은 단위로 나누는 것입니다
바로 선과 직사각형으로 나누는 것이지요

English: 
- [Voiceover] We've used the definite
integral to find areas.
What I want to do now is to see if we can
use the definitive role
to find an arc length.
What do I mean by that?
Well, if I start at this point
on the graph of a function,
and if I were to go at
this point right over here,
not a straight line, we
know already how to find
the distance in the straight
line but instead we want
to find the distance along the curve.
If we lay a string along the curve,
what would be the
distance right over here?
That's what I'm talking
about by arc length.
What we could think about it
is okay, that's going to be
from x equals a to x
equals b along this curve.
So how can we do it?
Well, the one thing that
integration, integral calculus is
teaching us is that when we see something
that's changing like
this, what we could do is
we can break it up into
infinitely small parts.
Infinitely small parts
that we can approximately

Portuguese: 
Usamos a integral definida
para calcular áreas.
Quero ver agora se conseguimos
usar a integral definida
para calcular um comprimento de arco.
O que quero dizer com isso?
Se começo neste ponto na função,
e vou até este outro ponto,
- não é uma reta; já sabemos calcular
a distância em um reta, mas agora
queremos achar a distância na curva. -
Se colocarmos uma corda na curva,
qual seria a distância até aqui?
É o que quero dizer
com comprimento de arco.
Podemos pensar que essa distância
vai de x igual à a até x igual à b.
Como podemos calcular isso?
O que o Cálculo Integral
nos ensinou é que quando vemos
alguma coisa variando assim,
podemos quebrá-la
em partes infinitesimais.
Partes infinitesimais
que podemos aproximar

Thai: 
เราได้ใช้อินทิกรัล
จำกัดเขตเพื่อหาพื้นที่ไป
สิ่งที่ผมอยากทำตอนนี้คือดูว่าเราใช้
อินทิกรัลจำกัดเขตหาความยาวส่วนโค้งได้ไหม
ผมหมายความว่าอะไร?
ถ้าผมเริ่มที่จุดนี้บนกราฟของฟังก์ชัน
และถ้าผมไปยังจุดนี่ตรงนี้
ไม่ใช่เส้นตรง เรารู้วิธีหา
ระยะตามเส้นตรง แต่เราอยาก
หาระยะตามเส้นโค้งแทน
ถ้าเราลากเส้นเชือกไปตามเส้นโค้ง
ระยะนี่ตรงนี้จะเป็นเท่าใด?
นั่นคือสิงที่ผมหมายถึงเวลาพูดว่า 
ความยาวส่วนโค้ง
สิ่งที่่เราคิดได้คือว่า โอเค มันจะเป็น
จาก x เท่ากับ a ถึง x เท่ากับ b ตามเส้นโค้งนี้
เราทำมันได้อย่างไร?
สิ่งหนึ่งที่การอินทิเกรต แคลคูลัสเชิงปริพันธ์
สอนเราคือว่า เวลาเราเห็นอะไร
ที่เปลี่ยนแปลงแบบนี้ สิ่งที่เราทำได้คือ
เราแยกมันเป็นส่วนเล็กจิ๋วได้
ส่วนเล็กจิ๋วที่เราสามารถประมาณ

Bulgarian: 
Използвахме определени
интеграли за намиране на площ.
Сега искам да видим 
дали можем
да използваме определен интеграл, 
за да намерим дължина на крива.
Какво имам предвид?
Ако започнем от тази
част на графиката на функцията,
и ако отидем до тази 
точка ето тук,
това не е права линия,
ние знаем дължината
на разстоянието по
права линия, но ако искаме
да намерим разстоянието
по протежение на кривата.
Ако сложим конец
върху кривата,
колко ще бъде разстоянието
от тук до тук?
Ето това имам предвид
под дължина на крива.
Тук можем 
да кажем, че ще бъде
от х = а до х = b
по самата крива.
Как да го направим?
Единият начин е интегриране.
Интегралното смятане
ни казва, че когато
видим нещо, което
се променя по този начин,
ние можем да го разделим
на безкрайно малки части.
Безкрайно малки части,
които можем приблизително

English: 
with things like lines and
rectangles, and then we could
take the infinite sum of
those infinitely small parts.
So let me break up my arc length.
Let me break it up into infinitely
small sections of arc length.
Let me call each of those
infinitely small sections
of my arc length a, I
guess I could say a length
to differential, an arc length
to differential, I call it ds.
I'll draw it a much bigger
that when I at least
I conceptualize what a differential is,
just so that we could see it.
What do I mean by breaking
it up into these ds's?
Well, if that's the ds,
and then let me do the
others in other colors,
that's another infinitely
small change in my arc length.
Another infinitely small
change in my arc length.
If I summed all of these ds's together,
I'm going to get the arc length.
The arc length, if I take
is going to be the integral
of all of these ds's sum
together over this integral

Korean: 
그리고 이 무한히 작은 부분들을
무한이 더하는 것입니다
그럼 이 곡선을 분리해 보겠습니다
무한히 작은 호로 나누어 봅시다
이 무한히 작은 길이를 ds라 부르겠습니다
눈으로 볼 수 있고 쉽게 이해하기 위해 그림은
훨씬 크게 그리겠습니다
이 선을 ds로 쪼갠 이유는 무엇일까요?
이것은 ds입니다
그리고 다른 색으로 다른 무한히 작은
호 길이의 변화를 표시하겠습니다
아 모든 ds값들을 더하면
저는 전체 호의 길이를 구할 수 있습니다
이 값은 모둔 ds의 합을 적분시킨 값이 됩니다
이 구간의 값을 적분시킵니다

Thai: 
ด้วยของอย่างเส้นตรงกับ
สี่เหลี่ยมมุมฉาก แล้วเราก็
หาผลบวกส่วนเล็กจิ๋วจำนวนนับไม่ถ้วนเหล่านั้น
ขอผมแบ่งความยาวส่วนโค้งนั่นนะ
ขอผมแบ่งส่วนโค้งนั้นเป็น
ส่วนเล็กๆ นับไม่ถ้วน
ขอผมเรียกส่วนเล็กๆ นับไม่ถ้วนนั้น
ว่าความยาวเส้นโค้ง ผมจะเรียกความยาว
ดิฟเฟอเรนเชียล ดิฟเฟอเรนเชียลของส่วนโค้ง
ผมจะเรียกมันว่า ds
ผมจะวาดมันใหญ่กว่า
สิ่งที่ผมคิดว่า ดิฟเฟอเรนเชียลควรเป็นอย่างไร
เราจะได้เห็นมันได้
การแบ่ง ds เหล่านี้หมายความว่าอะไร?
ถ้านั่นคือ ds
แล้วขอผมทำอันอื่นด้วยสีอื่นนะ
มันมีการเปลี่ยนแปลง
ความยาวส่วนโค้งเล็กจิ๋วอื่นๆ
การเปลี่ยนแปลงความยาวโค้งเล็กๆ อื่นๆ
ถ้าผมบวก ds ทั้งหมดนี้เข้าด้วยกัน
ผมจะได้ความยาวส่วนโค้ง
ความยาวส่วนโค้ง ถ้าผมหา จะได้อินทิกรัล
ของ ds เหล่านี้ทั้งหมดรวมกัน ทั้งอินทิกรัลนี้

Portuguese: 
com retas e retângulos, e depois
podemos somar essas partes.
Deixe-me dividir
o meu comprimento de arco.
Deixe-me quebrá-lo
em partes infinitesimais
de comprimento de arco.
Chamarei essas partes infinitesimais
do meu comprimento de arco de -
Chamarei de ds.
Farei o desenho em uma escala maior
do que seria um diferencial.
Apenas para enxergá-lo.
O que quero dizer dividindo isso
em vários ds?
Se isto é ds -
deixe-me usar outras cores-
aquilo é uma variação infinitesimal
no meu comprimento de arco.
Outra variação infinitesimal
no meu comprimento de arco.
Se eu somar todos esses ds,
obterei o comprimento de arco.
O comprimento de arco será a integral
da soma de todos esses ds -

Bulgarian: 
да оприличим на отсечки или
правоъгълници и после можем
да намерим сбора на безкрайно
голям брой такива безкрайно малки части.
Хайде да раздробим
тази дължина на кривата.
Да я разделим на безкрайно
малки части от дължината на кривата.
Ще нарека всяко от тези
безкрайно малки парченца
от дължината на кривата, 
предполагам, че мога да ги нарека
дължина към диференциал,
ще ги означа с ds.
Ще нарисувам това по-едро,
когато обясня какво е диференциал,
за да можеш да го виждаш.
Какво имам предвид под
раздробяване на такива ds?
Ако това е ds,
ще направя другите в
различни цветове,
това е друго безкрайно малко 
парченце от дължината на кривата.
Друга безкрайно малка
промяна в дължината на кривата.
Ако събера всички тези ds,
ще получа дължината
на кривата.
Дължината на кривата
ще бъде интеграл
от всички тези ds, сумирани 
в границите на интеграла,

Thai: 
เราก็เขียนมันแบบนี้ได้
แต่มันไม่ช่วยผมนักตอนนี้
นี่คือเทอมของความยาวส่วนโค้ง
ที่เป็นดิฟเฟอเรนเชียล
เรารู้วิธีทำสิ่งต่างๆ ในรูปของ dx กับ dy
ลองดูว่าเราเขียนตัวนี้ใหม่
ในรูปของ dx กับ dy ได้ไหม
ถ้าเราไปยังระดับที่เล็กจริงๆ
เหมือนเดิม เราประมาณได้
อันนี้จะเป็นเส้นตรง
วิธีที่เราประมาณ
พื้นที่เป็นสี่เหลี่ยมมุมฉากตอนแรก
แต่เรามีสี่เหลี่ยมเล็กจิ๋ว
จำนวนนับไม่ถ้วน เรา
กำลังประมาณเขตที่ไม่ใช่รูปสี่เหลี่ยมมุมฉาก
พื้นที่ของเขตที่ไม่ใช่สี่เหลี่ยมมุมฉาก
เช่นเดียวกัน เราประมาณเส้นตรง
เล็กจิ๋ว และมีจำนวนเส้นนับไม่ถ้วน
เรากำลังหาความยาวส่วนโค้งนี้
ตอนนี้สนใจแค่ส่วนของเส้นตรงนี้ก่อน
ระยะนี่ตรงนี้ ผมจะเขียน
มันในรูปของ dx กับ dy
ระยะนี่ตรงนี้ นั่นคือ dx
คุณทำอย่างนี้ได้เมื่อ
การเปลี่ยนปลงเล็กจิ๋วของ x
และระยะนี่ตรงนี้ นี่คือ dy

Bulgarian: 
което можем да запишем
ето така.
Но това още не ми помага.
Това е по отношение на диференциала
от дължината на кривата.
Ние знаем как да работим
с dy и dx.
Да видим как можем
да изразим това
чрез dx и dy.
Ако увеличим това много пъти,
можем да използваме
приближение.
Това тук ще бъде права.
По същия начин приблизително
определяхме площи отначало.
Но ако имаме безкраен брой
безкрайно малки 
правоъгълници, тогава
можем да изчислим приблизително
област, която не е правоъгълна.
Площта на област, която
не е правоъгълна.
Подобно, ние имаме
прави линии, които
са безкрайно малки и имаме 
безкрайно голям брой от тях,
като така намираме дължината
на кривата.
Да се фокусираме върху
тази права за момента.
Това разстояние тук ще се опитам 
да изразя чрез dy и dx.
Това разстояние тук е dx.
Това е безкрайно малка
промяна на х,
а това разстояние тук е dу.

Portuguese: 
podemos denotá-lo assim.
Isso não me ajudará agora.
Isto está em termos do cumprimento
de arco, que é um diferencial.
Sabemos calcular em termos de dx ou dy.
Vejamos se podemos reescrever isso
em termos de dx e de dy.
Se estivermos numa escala muito pequena
podemos fazer algumas aproximações.
Isto será uma reta.
Mesmo princípio que usamos
para aproximar áreas
com retângulos.
Mas se temos
um número infinito
de retângulos infinitesimais,
estamos aproximando
uma região não retangular.
A área de uma região não retangular.
Da mesma forma
que aproximamos com retas.
Há um número infinito delas,
então encontraremos o comprimento do arco.
Por enquanto, isto é uma reta.
Esta distância - tentarei
expressá-la em termos de dx e de dy-
é dx.
Você pode ver isso como uma variação
infinitesimal em x.
Esta distância é dy.

English: 
so we can denote it like this.
But this doesn't help me right now.
This is in terms of this arc
length that's differential.
We do know how to do things
in terms of dx's and dy's.
Let's see if we can re-express this
in terms of dx's and dy's.
If we go on a really, really small scale,
once again, we can approximate.
This is going to be a line.
We just the way that we approximated
area with rectangles at first.
But if we have an infinite number
of infinite small
rectangles, we're actually
approximating a non-rectangular region.
The area on a non-rectangular region.
Similarly, we're approximating with lines
with the infinitely small
and there's infinite number
of them, you are actually
finding the length of the curve.
Well, just focusing on
this is a line for now.
This distance right over
here, I'm just going
to try express in terms of dx's and dy's.
So this distance right
over here, that's dx.
You can do this as
infinitely small change in x,
and this distance right
over here, this is a dy.

Korean: 
이 식과 같이 적을 수 있습니다
하지만 이는 지금 당장 도움이 되지 않습니다
이는 호의 차이를 표시한 식이며
이를 dx와 dy로 표현하는 방법을
우리는 알 고 있습니다
그럼 dx와 dy의 값으로 다시 표현해 보겠습니다
가능한 작은 값으로 쪼갠다면
거의 직선에 가까울 것입니다
대략적인 직사각형의 넓이를
계산했던 방법 말이지요
무한이 많은 무한히 작은
직사각형을 이용해서
직사각형이 아닌 넓이를 구하는 것이지요
직사각형이 아닌 넓이요
동일한 방법으로 우리는 호의 길이를
무한히 작고 무한지 작은 길이의 
직선으로 계산해서
결국 전체 호의 길이를 계산하는 것이지요
이제 이 선에 집중해 보겠습니다.
dx와 dy로 표현해 보겠습니다
이 가로길이는 dx입니다
x의 무한히 작은 변화량 값이며
세로 길이는 dy입니다

English: 
Once again, I'm being
loosey-goosey with differentials.
Really giving you
conceptual understanding,
not a reader's proof, but
it'll give you a sense
of where the formula for arc
length is actually coming from.
Based on this, you can see
the ds could be expressed
based on the Pythagorean
Theorem as equal to dx squared
plus dy squared, or you could rewrite this
as square root of dx
squared plus dy squared.
So we could rewrite this.
We could say this is the same
thing as the integral of,
instead of writing ds,
I'm going to write it
as the square root of dx
squared plus dy squared.
Once again, this is straight
out of the Pythagorean Theorem.
Now this is starting to get interesting.
I've written in terms of dx's and dy's
but they're getting squared.
They're under radical sign.
What can I do to simplify this?

Korean: 
이를 활용하여 다시 한 번
이 길이를 계산해 보겠습니다
개념적으로 이 호의 길이가 어떻게
구해질 수 있는지 생각해보겠습니다
이 삼각형에서 피타고라스의 정리를 활용해서
ds의 길이를 구할 수 있습니다
이는 dx의 제곱과 dy의 제곱을 더한 값이
ds의 제곱과 같습니다
또한, ds는 dx제곱과 dy의 제곱을 더한 값의
제곱근과 같다고 할 수 있습니다
ds적분의 식을 다시 적어보겠습니다
이 식을 다시 적어보면 ds의 적분이며
이를 정리하면 √(dx^2+dy^2) 을
적분시킨 값과 같습니다
이 값은 피타고라스의 이론을 이용한 것입니다
여기에서 재미있는 것은 dx와 dy의 제곱의
값들로 표현할 수 있다는 것입니다
이 식을 어떻게 간단하게 표현할 수 있을까요?

Portuguese: 
Mais uma vez, não estou sendo
rigoroso com os diferenciais.
Apenas uma noção conceitual.
Não é uma prova rigorosa,
mas dará uma noção
de onde vem a fórmula
de comprimento de arco.
Com base nisso, você pode ver
que ds pode ser escrito,
seguindo o Teorema de Pitágoras,
como dx ao quadrado
mais dy ao quadrado.
Ou, você pode reescrever isso
como a raiz de dx ao quadrado
mais dy ao quadrado.
Podemos reescrever isto.
Podemos dizer que isto
é o mesmo que a integral -
em vez de escrever ds, escreverei
como a raiz quadrada de dx ao quadrado
mais dy ao quadrado.
Mais uma vez, isto veio
do Teorema de Pitágoras.
Isto está começando a ficar interessante.
Escrevi em termos de dx e dy,
mas eles estão ao quadrado.
Estão dentro de uma raiz.
O que posso fazer para simplificar isto?

Thai: 
ย้ำอีกครั้ง ผมไม่ได้ทำอะไรรัดกุม
กับดิฟเฟอเรนเชียลนัก
แค่ให้คุณเข้าใจหลักการ
ไม่ใช่การพิสูจน์ให้ดู แต่มันทำให้คุณพอเข้าใจ
ว่าสูตรของความยาวส่วนโค้งมาจากไหน
จากอันนี้ คุณเห็นได้ว่า ds เขียนได้
จากทฤษฎีบทพีทากอรัสว่า
เท่ากับ dx กำลังสอง
บวก dy กำลังสอง หรือคุณเขียนใหม่ได้
เป็นรากที่สองของ dx กำลังสอง
บวก dy กำลังสอง
เราก็เขียนอันนี้ใหม่ได้
เราบอกได้ว่า อันนี้เท่ากับอินทิกรัลของ
แทนที่จะเขียน ds ผมจะเขียนมัน
เป็นรากที่สองของ dx กำลังสอง
บวก dy กำลังสอง
เหมือนเดิม อันนี้ตรงมาจาก
ทฤษฎีบทพีทากอรัส
ตอนนี้ มันเริ่มดูน่าสนใจแล้ว
ผมได้เขียนมันในรูปของ dx กับ dy
แต่พวกมันกำลังสองอยู่
พวกมันอยู่ใต้เครื่องหมายราก
ผมจัดรูปอันนี้ได้อย่างไร?

Bulgarian: 
Аз знам добре какво да правя
с тези диференциали.
Но искам да добиеш представа,
това не е доказателство,
просто ти давам логиката
откъде идва формулата
за дължината на крива.
Въз основа на това можеш
да видиш, че ds е изразено
чрез питагоровата теорема
като равно на dx^2 + dy^2.
което можем да преработим като
корен квадратен от 
(dx^2 + dy^2).
Това можем да го преработим.
Това е равно на интеграл от,
но вместо ds,
ще запиша това като
квадратен корен от (dx^2 + dy^2).
Това следва директно
от питагоровата теорема.
И сега започва
да става интересно.
Представихме го чрез dx и dy,
но те са на квадрат.
Те са под знак за корен.
Как да опростим това?

Portuguese: 
Ou, pelo menos, escrever de uma forma
que eu saiba integrar.
Posso fatorar um dx ao quadrado.
Deixe-me reescrever.
Isto será o mesmo
que a integral da raiz quadrada-
vou fatorar o dx ao quadrado-
dx ao quadrado vezes um
mais dy sobre dx ao quadrado.
Veja que esses dois são iguais.
Se eu distribuir este dx ao quadrado,
obterei isto aqui.
Agora posso colocar o dx ao quadrado
para fora da raiz.
Isto será a integral de-
deixe-me escrever isto em branco-
a integral de um mais dy
sobre dx ao quadrado.
Isto é interessante, pois sabemos
o que dy sobre dx é.
É a derivada da nossa função,
dy sobre dx, ao quadradado
Se fatorar dx ao quadrado,

Thai: 
หรืออย่างน้อย เขียนมันในรูป
ที่ผมรู้วิธีการอินทิเกรต
ผมแยก dx กำลังสองออกมาได้
ขอผมเขียนมันใหม่นะ
อันนี้จะเท่ากับ
อินทิกรัลของรากที่สอง
ผมจะแยก dx กำลังสองออกมา
dx กำลังสองคูณ 1 บวก dy ส่วน dx กำลังสอง
สังเกตว่าอันนี้ อันนี้คือปริมาณเดียวกัน
ถ้าผมแจกแจง dx กำลังสองนี้
ผมจะได้ส่วนนี่ตรงนี้
ทีนี้ ผมเอา dx กำลังสองออกจากรากได้
อันนี้จะเท่ากับอินทิกรัลของ
ขอผมเขียนด้วยสีขาวนะ
อินทิกรัลของ 1 บวก dy/dx กำลังสอง
อันนี้น่าสนใจเพราะเรารู้ว่า dy/dx คืออะไร
นี่คืออนุพันธ์ของฟังก์ชัน
dy/dx กำลังสอง
ถ้าคุณนำ dx กำลังสองออกจากราก

English: 
Or at least write it in a way
that I know how to integrate.
Well, I could factor out a dx squared,
so let me just rewrite it.
This is going to be the same thing
as the integral of the square root.
I'm going to factor out the dx squared,
dx squared times one
plus dy over dx squared.
Notice this, and this
is exact same quantity.
If I distribute this dx squared,
I'm going to get this right up here.
Now I can take the dx
squared out of the radical.
So this is going to be the integral of,
let me write that in the white color,
the integral of one plus dy, dx squared.
This is interesting because
we know what dy, dx is.
This is the derivative of our function,
dy, dx squared.
If you take the dx squared
out of the radical,

Korean: 
아니면 적어도 우리가 적분을 할 수 있는
식으로 표현해 봅시다
이 식에서 dx제곱근을 꺼내도록 하겠습니다
이를 다시 적어보겠습니다
다음의 식과 같이 됩니다
√dx^2{1+(dy/dx)^2}의 적분 값
이 두 식은 정확이 같습니다
dx^2을 다시 괄호안이 식에 곱하면
정확히 동일한 식이되겠지요?
이번엔 dx를 루트기호(√)밖으로 꺼내겠습니다
식은 다음과 같습니다
흰색으로 그리겠습니다
√1+(dy/dx)^2 의 적분값으로 정리됩니다
이는 매우 흥미롭습니다
dy/dx가 그래프를 나타내는 함수의
적분값이기 때문이지요
(dy/dx)에 제곱을 표시하겠습니다

Bulgarian: 
Или поне да го напишем така,
че да можем да го интегрираме.
Тук мога да изнеса 
пред скоби dx^2,
само да препиша това.
Това ще бъде равно на
интеграл от квадратен корен.
Ще изнеса пред скоби dx^2,
dx^2 по (1 + dy^2/dx^2).
Обърни внимание, че
тези двете са равни.
Ако умножа по dx^2,
ще получа отново този израз.
И сега мога да изнеса dx
от знака за корен.
Това ще е равно на
интеграл от...
ще го запиша с бяло:
интеграл от 1 + (dy/dx)^2.
Това е интересно, защото ние знаем 
какво представлява dy/dx.
Това е производната на функцията,
(dy/dx)^2.
Ако изнесем dx^2
извън знака за корен,

Bulgarian: 
корен квадратен от dx^2
е просто dx.
Това е просто dx.
И сега става още по-интересно,
защото знаем как
да решим това между две граници.
Можем да решим определен
интеграл от а до b.
Сега интегрираме куп dx-ове,
или интегрираме спрямо х.
Можем да кажем: 
"х е равно на а и х е равно на b".
Хайде да съберем
произведението
на този израз и dx,
а това е важно.
Това е формулата
за дължината на равнинна крива.
Изглежда сложна.
В следващото видео
ще видим, че всъщност
тя много лесно се използва,
макар понякога сметките
да стават сложни.
Ако искаш да запишеш това
по малко по-различен начин,
можеш да запишеш, че това
е равно на интеграл от а до b,
ат x = a до x = b от корен 
квадратен от 1 плюс...
вместо dy, dx, мога
да запиша тук

English: 
the square root of dx squared
is just going to be dx.
This is just going to be dx.
This is just going to be dx.
Now this is really
interesting because we know
how to find this between two bounds.
We can now take the
definitive role from a to b.
This is now we are
integrating a bunch of dx's
or we're integrating with respect to x.
We could say, "Okay, x
equals a to x equals b."
Let's take the sum of the product
of this expression and
dx, and this is essential.
This is the formula for arc length.
The formula for arc length.
This looks complicated.
In the next video, we'll
see there's actually
fairly straight forward to apply
although sometimes in math gets airy.
If you wanted to write this in
slightly different notation,
you could write this as equal
to the integral from a to b,
x equals a to x equals b of
the square root of one plus.
Instead of dy, dx, I could write it
as f prime of x squared, dx.

Portuguese: 
a raiz quadrada de dx ao quadrado
será apenas dx.
Isto será dx.
Será dx.
Isto é interessante,
pois sabemos
calcular esta diferença.
Agora podemos integrar de a até b.
Estamos integrando vários dx,
ou estamos integrando em relação à x.
Podemos dizer de x igual à a
até x igual à b.
Vamos calcular a soma do produto
desta expressão e dx.
Isto é a fórmula de comprimento de arco.
Parece complicado.
No próximo vídeo, veremos
que há uma aplicação direta,
mas os cálculos podem
complicar-se um pouco.
Se quiser usar uma notação diferente,
você escrever como a integral de a até b.
x igual à a até x igual à b
da raiz quadrada de um mais -
em vez de dx sobre dy,
posso colocar

Korean: 
dx를 루트기호(√)밖으로 빼내면
dx가 됩니다
우리는 정해진 구간 안에서 
이 값을 구할 수 있습니다
a에서 b사이 구간의 값을 구해보겠습니다
dx로 적분을 시키겠습니다
x를 중심으로 적분을 시키겠습니다
x가 a에서 b까지의 구간이며
그럼 관호안의 식과 dx를 적분해 보겠습니다
이 것은 바로 호를 나타내는 함수이지요
약간 복잡해보이는데요
다음영상에서 정확히 어떻게
적분을 입하는지 배우겠습니다
약간 다른표기법으로 작성하고 싶다면
이는 다음의 식과 같이 쓸 수 있습니다
x가 a에서 b까지의 구간에서

Thai: 
รากที่สองของ dx กำลังสองจะเท่ากับ dx
อันนี้จะเท่ากับ dx
อันนี้จะเท่ากับ dx
อันนี้น่าสนใจจริงๆ เพราะเรารู้
วิธีหาค่านี้ระหว่างขอบสองตัว
ตอนนี้เราหาปฏิยานุพันธ์จาก a ถึง b ได้
นี่คือ เรากำลังอินทิเกรต dx หลายๆ ตัว
หรือเราอินทิเกรตเทียบกับ x
เราบอกได้ว่า โอเค x เท่ากับ a ถึง x เท่ากับ b
ลองหาผลบวกของผลคูณ
ของพจน์นี้ กับ dx และอันนี้สำคัญ
นี่คือสูตรความยาวส่วนโค้ง
สูตรสำหรับความยาวส่วนโค้ง
อันนี้ดูซับซ้อน
ในวิดีโอหน้าเราจเห็นว่ามัน
ใช้ได้ค่อนข้างตรงไปตรงมา
ถึงแม้ว่าบางครั้งเลขอาจจะดูยาก
ถ้าคุณอยากเขียนอันนี้อีกแบบ
คุณก็เขียนได้ว่า 
อันนี้เท่ากับอินทิกรัลจาก a ถึง b
x เท่ากับ a ถึง x เท่ากับ b 
ของรากที่สองของ 1 บวก
แทนที่จะเป็น dy/dx ผมเขียนมัน
เป็น f ไพรม์ของ x กำลังสอง, dx

Korean: 
{√1+f'(x)^2}dx
자 그럼 f(x)의 함수를 알고 있다면
x를 기준으로 미분을 시키고
1을 더하고 그 값의 제곱근을 구해
그 값을 x를 기준으로 적분시켜 a에서 b구간
값을 구하면 되며 다음 영상에서 이어가겠습니다

English: 
So if you know the function,
if you know what f of x is,
take the derivative of it
with respect to x squared
added to one, take the square
root, and then multiply,
and then take the definite integral
of that with respect to x from a to b.
We'll do that in the next video.

Bulgarian: 
(f'(x)^2)dx.
Така че ако знаеш колко е f(х),
намираш производната
спрямо (х^2 + 1),
коренуваш и после
умножаваш,
и след това намираш
определен интеграл
от това спрямо х
в интервала от а до b.
Ще го направим в
следващото видео.

Thai: 
ถ้าเรารู้ฟังก์ชัน ถ้าเรารู้ว่า f ของ x คืออะไร
หาอนุพันธ์ของมันเทียบกับ x กำลังสอง
บวก 1 แล้วหารากที่สอง แล้วคูณ
แล้วหาอินทิกรัลจำกัดเขต
ของมันเทียบกับ x จาก a ถึง b
เราจะทำมันในวิดีโอหน้านะ

Portuguese: 
f linha de x ao quadrado, dx.
Se você sabe qual é a função, f de x,
calcule a derivada em relação à x,
ao quadrado mais um,
calcule a raiz quadrada e multiplique.
E depois, calcule a integral definida
em relação à x, de a até b.
Faremos isso no próximo vídeo.
Legendado por: [Pilar Dib]
Revisado por [Victor Oliveira]
