
English: 
Let's do some more volumes
of solids of revolutions.
So let's say that
I have the graph
y is equal to square root of x.
So let's do it, so it
looks something like this.
So that right over there is y is
equal to the square root of x.
And let's say I also have
the graph of y equals x.
So let's say y equals x
looks something like this.
It looks just like that.
y equals x.
And what I care about
now is the solid
I get if I were to rotate the
area between these two things
around to the x-axis.
So let's try to visualize it.
So the outside is going to
be kind of a truffle shape,
it's going to look
like a truffle shape.
And then we hollow out
a cone inside of it.
So I'll make my best
attempt to draw this shape.

Korean: 
회전체의 부피를 구하는 
활동을 계속해 봅시다
회전체의 부피를 구하는 
활동을 계속해 봅시다
y=√x의
그래프가 있다고 합시다
이처럼 생겼습니다
여기 y=√x의 그래프가 있습니다
또, y=x의 그래프도 있다고 합시다
y=x의 그래프는 다음과 같이 생겼습니다
이렇게 생겼습니다
y=x
이 두 그래프 사이의 영역을
x 축에 대하여 회전할 경우 생기는
회전체에 대해 생각해봅시다
그려 봅시다
바깥쪽은 트뤼플 모양일 것이고
트뤼플처럼 생겼을 것입니다
안쪽은 원뿔처럼 파여있을 것입니다
잘 그리려고 노력해보겠습니다

Bulgarian: 
 
Да изчислим обемите
на още няколко ротационни тела.
Да кажем, че имаме графиката
на у е равно на квадратен корен от х.
Значи графиката 
ще изглежда ето така.
Значи това тук е у равно на
квадратен корен от х.
Нека имаме графиката на
у на х.
Да кажем, че у = х е нещо такова.
То изглежда ето така.
у = х.
Какво тяло ще получа
при ротация на площта между
тези две графики
около оста х.
Да го визуализираме.
Това ще бъде нещо като
чаша или трюфел,
И после издълбаваме вътре
един конус.
Ще се опитам да го 
нарисувам колкото мога по-добре.

Portuguese: 
Vamos fazer mais alguns volumes de
sólidos de revoluções.
Digamos que eu tenha o gráfico
y é igual a raíz quadrada de x.
Vamos fazer. Se parece com algo assim.
Então isso aqui é y igual a
raíz quadrada de x.
E vamos dizer que também tenho
o gráfico de y igual a x.
Digamos que y igual a x
parece com algo assim.
Algo parecido com isso.
y igual a x.
E me preocupo agora com o sólido
que obtenho se eu rotacionar
a área entre essas duas coisas
ao redor do eixo x.
Vamos tentar visualizar.
A parte externa terá uma forma de trufa,
parecida com um formato de trufa.
E então fazemos um buraco
em forma de cone dentro dele.
Farei meu melhor pra desenhar essa figura.

Thai: 
 
ลองหาปริมาตรทรงตันจากการหมุนอีกกัน
สมมุติว่าเรามีกราฟ
y เท่ากับรากที่สองของ x
ลองทำดู มันจะเป็นแบบนี้
เส้นนี่ตรงนี้คือ y เท่ากับรากที่สองของ x
และสมมุติว่าผมมีกราฟของ y เท่ากับ x ด้วย
สมมุติว่า y เท่ากับ x เป็นแบบนี้
มันเป็นอย่างนั้น
y เท่ากับ x
และสิ่งที่ผมสนใจคือทรงตัน
ที่ผมได้เมื่อหมุนพื้นที่ระหว่างสองตัวนี้
รอบแกน x
ลองมองภาพกัน
ข้างนอกจะเป็นทรงรูปขนมทรัฟเฟิล
มันดูเหมือนรูปขนมทรัฟเฟิล
แล้วเราคว้านกรวยข้างในออกมา
ผมจะพยายามวาดรูปนี้ให้ดีที่สุดนะ

Bulgarian: 
Ще изглежда горе-долу така –
отвън ще изглежда ето така.
Ще изглежда ето така.
Интересува ни интервалът.
Интересува ни интервалът
между пресечните им точки.
Между тази точка
и тази точка тук.
Външната част ще изглежда
ето така.
Това е основата на чашата.
Това е основата на чашата 
или на трюфела,
тялото ще прилича
на трюфел отвън.
Но вероятно сме на 
някаква диета,
не искаме да изядем
целия трюфел.
Затова издълбаваме 
конус вътре.
Вътрешността
практически е куха,
прилича на черупка.
Изрязваме в центъра конус.
Правим ротация около оста х.
Трюфел отвън, в който
е издълбан конус отвътре.
Какъв е обемът на
това тяло?

Portuguese: 
Será parecido com algo-- a parte externa
será parecida com algo assim.
Vai ser mais ou menos assim.
Nós nos importamos com o intervalo.
Nos importamos com o intervalo entre
os pontos que eles interceptam.
Entre esse ponto e esse ponto aqui.
A parte externa será parecida
com algo assim.
Essa é a base da trufa.
Essa é a base da trufa,
terá esse formato de trufa
na parte externa.
Acho que talvez estamos
em algum tipo de dieta,
não queremos comer a trufa inteira.
Então cavamos um cone no interior.
Então seu interior é essencialmente
oco exceto por essa parte de casca.
Então cavamos um cone no centro.
Rotacionamos ao redor do eixo x.
Trufa do lado de fora, um cone escavado
no lado de dentro.
E qual será o volume disso?

Korean: 
이렇게 생겼을 것입니다
바깥쪽은 이렇게 생겼을 것이고
이렇게 생겼을 것인데
중요한 것은 범위입니다
두 그래프가 만나는 두 점 사이의
범위에 해당하는 점들에 관심이 있습니다
이 점과 이 점 사이 말이지요
바깥쪽은 이렇게 생겼을 것이고
트뤼플 모양의 밑입니다
이게 트뤼플 모양의 밑부분입니다
바깥쪽에는 이처럼 
트뤼플 모양을 가질 것입니다
그런데, 다이어트를 하고 있다면
트뤼플을 전부 다 먹고 싶지는 않겠죠
안쪽에 원뿔 모양으로 
트뤼플을 파내어 봅시다
따라서, 껍질 같은 부분 빼고
안쪽은 거의 텅 비어 있는 것이지요
안쪽에 원뿔 모양으로 트뤼플을 파내어 봅시다
위 그래프 사이의 영역을 x 축에 대해 회전하면
바깥쪽은 트뤼플 모양
안쪽은 원뿔처럼 파여 있습니다
이 회전체의 부피는 어떠할까요

English: 
So it's going to look something
like-- so the outside is
going to look
something like this.
It's going to look
something like that.
And we care about the interval.
We care about the interval
between the points
that they intersect.
So between this point
and this point here.
So the outside is going to
look something like this.
So this is the base
of the truffle.
That is the base of
the truffle, it's
going to have this kind of
truffle shape on the outside.
But I guess maybe we're
on some type of a diet,
we don't want to eat
to the entire truffle.
So we carve out a
cone on the inside.
So the inside of
it is essentially
hollow except for this
kind of shell part.
So we carve out a
cone in the center.
So we rotate it
around to the x-axis.
Truffle on the outside, carved
out a cone on the inside.
So what's going to be
the volume of that thing?

Thai: 
มันจะเป็นแบบ -- ข้างนอก
จะเป็นแบบนี้
มันจะเป็นแบบนั้น
และเราสนใจช่วง
เราสนใจช่วงระหว่างจุด
ที่พวกมันตัดกัน
ระหว่างจุดนี้กับจุดนี้ตรงนี้
ข้างนอกจะเป็นแบบนี้
นี่คือฐานของทรัฟเฟิล
นี่คือฐานของทรัฟเฟิล มัน
จะมีรูปร่างเหมือนทรัฟเฟิลจากข้างนอก
แต่ผมว่าเรากำลังคุมอาหารอยู่
เราไม่อยากกินทรัฟเฟิลทั้งอัน
เราจึงคว้านเอากรวยข้างในออก
ข้างในมันกลวง
ยกเว้นส่วนที่เป็นเปลือก
เราคว้านกรวยตรงกลางออก
แล้วเราหมุนมันรอบแกน x
ทรัฟเฟิลข้างนอก คว้านกรวยข้างในออก
ปริมาตรของรูปนั้นจะเป็นเท่าใด?

Portuguese: 
Basicamente, se pegarmos
uma fatia da nossa figura,
essa parte será a parede.
E basicamente, vamos pegar
o volume dessa parede inteira
que estamos
rotacionando ao redor do eixo x.
E como fazemos isso?
Bem, pode parecer pra você que
se acharmos o volume da trufa
sem ter sido escavada,
e então subtrair disso o volume do cone,
basicamente, encontraríamos o volume
do espaço entre a parte externa da trufa
e a parte do cone da trufa.
E como faríamos isso?
Bem, para achar o volume
da parte externa--
deixe-me desenhar isso aqui.
Na verdade, vou desenhar aqui.
Se pensarmos sobre o volume
da parte externa,
novamente, podemos usar
o método do disco.
Então em um ponto qualquer no tempo
nosso raio para um de nossos discos
será igual a função.
Vamos rotacionar esse disco.
Vou fazer numa cor diferente.
É difícil ver o disco porque está
no mesmo magenta.

English: 
So it's essentially, if we
take a slice of our figure,
it's going to be, this
is going to be the wall.
And we're essentially
going to take
the volume of this
entire wall that we're
rotating around the x-axis.
So how do we do that?
Well, it might dawn
on you that if we
found the volume of the
truffle if it was not
carved out, and then
subtract from that the volume
of the cone, we
would essentially
find out the volume of the
space in between the outside
of the truffle and the
cone part of the truffle.
So how would we do that?
Well, so to find the
volume of the outer shape.
So let me draw it over here.
So actually let me
just draw it over here.
If we think about the
volume of the outer shape,
once again, we can
use the disk method.
So at any given point in time
our radius for one of our disks
is going to be equal
to the function.
Let's rotate that disk around.
Actually let me do it
in a different color,
it's hard to see that
disk because it's
in the same magenta.

Thai: 
มันก็คือ ถ้าเราตัดรูปออกมา
มันจะ อันนี้จะเป็นผนัง
เราจะนำ
ปริมาตรของผนังทั้งหมดนี้ที่เรา
หมุนรอบแกน x
เราจะทำได้อย่างไร?
คุณอาจเห็นว่า ถ้าเรา
รู้ปริมาตรทรัฟเฟิลตอนที่มันไม่ถูก
คว้านออก แล้วลบปริมาตร
กรวย เราก็จะ
หาปริมาตรที่ว่างระหว่างข้างนอก
ทรัฟเฟิลกับส่วนที่เป็นกรวยของทรัฟเฟิลได้
แล้วเราหามันได้อย่างไร?
เวลาหาปริมาตรของรูปนอก
ขอผมวาดมันตรงนี้นะ
ขอผมวาดมันตรงนี้นะ
ถ้าเราคิดถึงปริมาตรของรูปนอก
เหมือนเดิม เราใช้วิธีแบบจานได้
ที่จุดใดๆ รัศมีของจานอันหนึ่ง
จะเท่ากับฟังก์ชันนั้น
ลองหมุนจานไป
ขอผมใช้อีกสีนะ
มันเห็นจานได้ยากเพราะมัน
มีสีม่วงเหมือนกัน

Korean: 
회전체의 단면을 살펴보면
이러한 모양을 관찰할 수 있습니다
x 축에 대하여 회전하여 얻은 물체에서
이 벽의 전체 부피를
구해볼 것입니다
어떻게 구할 수 있을까요?
트뤼플 모양이 파여 있지 않을 경우의
부피를 구한 뒤
원뿔 모양의 부피를 빼면
바깥쪽 트뤼플 모양과
안쪽 원뿔 모양 사이 부분의
부피를 구할 수 있을 것입니다
어떻게 구할 수 있을까요?
바깥쪽 트뤼플 모양의 부피를 구해 봅시다
여기 그려 보겠습니다
여기 그려 보겠습니다
바깥쪽 트뤼플 모양의 부피를 생각해 보면
디스크 방법을 사용할 수 있을 것입니다
이 물체의 어느 점에서든
원판의 지름은
함숫값과 일치할 것입니다
원판을 회전시켜 봅시다
다른 색을 사용해 보겠습니다
같은 자주색이어서
원판을 식별하기 힘들군요

Bulgarian: 
Ако отрежем резен
от това тяло,
то ще бъде...
това ще бъде стената.
Практически ще изчислим
обема на цялата тази стена,
която е завъртяна около оста х.
Как ще го направим?
Може би ти хрумва, че ако
намерим обема на този 
трюфел, когато не е издълбан,
и после извадим обема 
на този конус,
ще получим обема на пространството
между външната и вътрешната
част на трюфела – между
външността и конуса.
Как ще го направим?
Да намерим първо обема на
външната част.
Ще я нарисувам ето тук.
Ще я направя тук всъщност.
Ако разгледаме обема
на външната част,
можем да я намерим като
обем на ротационно тяло.
Във всяка точка радиусът
на нашите дискове
е равен на тази функция.
Да завъртим този диск.
Ще използвам различен цвят,
защото е трудно
да видим диска, който
е в същия цикламен цвят.

English: 
So this is a radius, and
let's rotate it around.
So I'm rotating the disk around.
This is our face of the disk,
that's our face of the disk.
It's going to have
a depth of dx.
We've seen this multiple times.
It's going to have
a depth of dx.
So the volume of
this disk is going
to be our depth, dx, times
the area of the face.
The area of the face is going to
be pi times the radius squared.
The radius is going to
be equal to the value
of the outer function, in this
case the square root of x.
So it's going to be pi
times our radius squared,
which is pi times the
square root of x squared.
And so if we want
to find the volume
of the entire outer
thimble, or truffle,
or whatever we want to call it.
Before we even carve
out the center,
we just take a sum of a bunch
of these, a bunch of these disks
that we've created.
So that's one disk, we would
have another disk over here.
Another disk over here.

Bulgarian: 
Това е радиус, който
завъртаме наколо.
Завъртам диска.
Това е лицето на диска,
лицето на нашия диск.
Той има дебелина dx.
Виждали сме го много пъти.
Дебелината е dх.
Обемът на диска ще бъде
дебелината dх, по площта
на повърхността.
Площта на тази повърхност
е π по радиуса на квадрат.
Радиусът е равен на стойността
на външната стойност, която
в този случай е корен квадратен от х.
Това е π по квадрата на радиуса,
което е равно на π по квадрата на 
корен квадратен от х.
Ако искаме да намерим обема
на цялото тяло, на трюфела,
или както искаш го наричай,
преди още да издълбаем центъра,
просто взимаме сумата на тези
дискове, които си представяме.
Това е един диск, тук има
друг диск.
Тук има друг диск.

Thai: 
นี่ก็คือรัศมี แล้วลองหมุนมันไป
ผมจะหมุนจานไป
นี่คือหน้าของจาน นั่นคือหน้าของจาน
มันจะลึก dx
เราเห็นอันนี้มาหลายครั้งแล้ว
มันจะลึก dx
ปริมาตรของจานจะ
เป็นความลึก dx คูณพื้นที่หน้าตัด
พื้นที่ของหน้าจะเท่ากับพายคูณรัศมีกำลังสอง
รัศมีจะเท่ากับค่า
ของฟังก์ชันตัวนอก 
ในกรณีนี้คือรากที่สองของ x
มันจะเท่ากับพายคูณรัศมีของเรากำลังสอง
ซึ่งก็คือพายคูณรากที่สองของ x กำลังสอง
แล้วถ้าเราอยากหาปริมาตร
ของถ้วยข้างนอก หรือทรัฟเเฟิล
เราอยากเรียกอะไรก็ได้
ก่อนที่เราจะคว้านตรงกลาง
เราแค่ผลบวกของจานเหล่านี้ จานเหล่านี้
ที่เราสร้างขึ้น
นั่นคือหนึ่งจาน เรามีอีกจานตรงนี้
อีกจานตรงนี้

Portuguese: 
Então esse é o raio, e vamos
rotacionar ao redor.
Estou rotacionando o disco ao redor.
Essa é a face do nosso disco.
Terá uma profundidade de dx.
Vimos isso várias vezes.
Terá profundidade dx.
Então o volume do disco será
nossa profundidade, dx,
vezes a área da face.
A área da face será pi vezes
o raio ao quadrado.
O raio será igual ao valor
da função externa,
nesse caso, a raiz quadrada de x.
Então será pi vezes o raio ao quadrado,
que é pi vezes a raiz quadrada
de x ao quadrado.
E se quisermos achar o volume
do dedal externo inteiro, ou trufa,
ou o nome que quisermos.
Antes de escavarmos o centro,
tomamos a soma de alguns desses,
alguns desses discos que criamos.
Então esse é um disco, teríamos
outro disco por aqui. Outro disco aqui.

Korean: 
이게 반지름인데, 회전시켜 봅시다
원판을 회전시키고 있습니다
이것이 원판의 앞면입니다
dx의 두께를 가지고 있죠
여러 번 보았을 겁니다
dx의 두께를 가지고 있죠
이 원판의 부피는
면의 넓이와 dx를 곱한 값입니다
면의 넓이는 π 곱하기 반지름의 제곱입니다
반지름은 바깥쪽 경계의 함숫값으로
이 경우에는 √x입니다
면의 넓이는 π 곱하기 반지름의 제곱입니다
π 곱하기 (√x)²입니다
트뤼플 모양 전체의
부피를 알고 싶다면
트뤼플이든 뭐든 말이죠
안쪽을 파내기 전에
우리가 만든 이 원판들을
모두 더하면 됩니다
이게 하나의 원판이고, 여기 또 다른 원판이 있겠죠
원판 하나 더

Portuguese: 
Para cada x, temos outro disco.
Continuando, enquanto os x's ficam maiores
os discos têm raio cada vez maiores.
Então vamos somar todos esses discos.
E vamos tomar o limite quando
cada um desses discos
ficam infinitamente finos, e temos
um número infinito deles.
Mas temos que descobrir nossos
limites de integração.
Quais os limites de integração?
Quais são os dois pontos aqui
onde eles interceptam?
Bem, podemos dizer que essas duas
coisas são iguais.
Se você diz x é igual a
raiz quadrada de x,
quando x é igual a raiz quadrada de x?
Podemos elevar os
dois lados ao quadrado.
E quando x ao quadrado
se iguala a x?
Você pode, podemos deixar aqui.
Poderia resolver isso, há
múltiplas maneiras de se fazer.
Mas você poderia resolver esse tipo
apenas pensando.
Se x é igual a zero, x
ao quadrado é igual a x.
E você vê aqui no gráfico,
x é igual a zero.
E também, um ao quadrado é igual a um.
Um é igual a raiz quadrada de um.
Poderia fazer outras coisas.

Korean: 
모든 x값에 해당하는 원판이 존재합니다
x값이 증가함에 따라
원판의 반지름도 증가합니다
이제 모든 원판을 더해야 합니다
각각의 원판이 무한히 얇아질 수 있도록
극한 기호를 취하면 무한개의 원판이 존재하게 됩니다
여기서 적분의 양 끝값을 알아내어야 합니다
적분의 양 끝값이 무엇일까요?
두 그래프가 만나는 두 점은 어디일까요?
두 식이 서로 같다고
놓고 계산하면 됩니다
x=√x라고 놓으면
x=√x가 성립하는 x값은 무엇일까요?
양변을 제곱하면
x=x²이 성립하는 x값은 무엇일까요?
잠시 생각해볼까요
이 문제를 해결하기 위한
다양한 방법이 존재합니다
암산으로도 쉽게 나오는군요
x=0일 때 위 식이 성립합니다
그래프에서 볼 수 있듯이
x=0일 때 성립합니다
또한, x=1일 때도 위 식이 성립합니다
1=1²이기 때문이죠
다른 방법을 사용할 수도 있습니다

Thai: 
สำหรับแต่ละ x เรามีอีกจาน
และเมื่อเราไป เมื่อ x มากขึ้นเรื่อยๆ
จานจะมีรัศมีมากขึ้นเรื่อยๆ
เราก็บวกจานทั้งหมดนี้
เราจะหาลิมิตเมื่อจานแต่ละอัน
บางเฉียบ และเรามีจำนวนนับไม่ถ้วน
แต่เราต้องหาขอบเขตการอินทิเกรต
ขอบของการอินทิเกรตคืออะไร?
จุดสองจุดที่กราฟตัดกันมีจุดไหนบ้าง?
เราแค่ตั้งสองตัวนี้
ให้เท่ากัน
ถ้าเราบอกว่า x เท่ากับรากที่สองของ x
x เท่ากับรากที่สองของ x เมื่อใด?
คุณก็ยกกำลังสองทั้งสองข้างนี้ได้
คุณก็ถามว่า x กำลังสองเท่ากับ x เมื่อใด?
คุณก็ เราทิ้งมันไว้อย่างนั้น
คุณแก้อันนี้ได้ มันมี
วิธีแก้ได้หลายวิธี
แต่คุณแก้สมการนี้ได้แค่คิดดีๆ
ถ้า x เท่ากับ 0, x กำลังสองจะเท่ากับ x
และคุณเห็นว่าบนกราฟนี่ตรงนี้ x
เท่ากับ 0
แล้ว 1 กำลังสองเท่ากับ 1 ด้วย
1 เท่ากับรากที่สองของ 1
คุณทำได้หลายวิธี

Bulgarian: 
За всяко х имаме друг диск.
И така, докато х става
все по-голямо и по-голямо,
дисковете имат
все по-големи радиуси.
Събираме обемите 
на всички дискове.
Намираме границата, когато
всеки от тези дискове
става безкрайно тънък и имаме
безкрайно много от тях.
Но трябва да определим
границите за интегриране.
Какви са границите за интегриране?
Какви са тези две точки тук,
където те се пресичат?
Можем да приравним
тези двете функции.
Ако сложим х е равно
на квадратен корен от х
кога х е равно на 
квадратен корен от х?
Повдигаме на квадрат двете страни.
Кога х^2 е равно на х?
Можем да решим това и тук.
Можем да го решим, тъй като
има няколко начина да го направим.
Но можем да го решим,
като просто разсъждаваме.
Ако х = 0, то х^2 = х.
Значи тук на графиката
х е равно на 0.
1 на квадрат също е равно на 1.
1 е равно на 
корен квадратен от 1.
Може да пробваш и други начини.

English: 
For each x we have another disk.
And as we go, as x's
get larger and larger,
the disks have a larger
and larger radius.
So we're going to sum
up all of those disks.
And we're going to take the
limit as each of those disks
get infinitely thin, and we
have an infinite number of them.
But we have to figure out our
boundaries of integration.
So what are our
boundaries of integration?
What are the two points right
over here where they intersect?
Well we could just
set these two things
to be equal to each other.
If you just said x is
equal to square root of x,
when does x equal
the square root of x?
I mean you can square
both sides of this.
You could say when
does x squared equal x?
You could, well we
could keep it there.
You could kind of
solve this, there's
multiple ways you could do it.
But you could solve this kind
of just thinking about it.
If x is equal to 0, x
squared is equal to x.
And you see that on the
graph right over here. x
is equal to 0.
And also 1 squared
is equal to 1.
1 is equal to the
square root of 1.
You could have
done other things.

Portuguese: 
Poderia dizer OK, x ao quadrado
menos x é igual a zero.
Poderia fatorar o x.
E temos x vezes x menos um
é igual a zero.
E ambos poderiam ser igual a zero,
então x é igual a zero ou
x menos um é igual a zero.
E temos x é igual a zero ou
x é igual a um.
O que nos dá os limites de integração.
Isso vai de x igual a zero até
x igual a um.
E agora para o lado de fora
podemos descobrir o volume.
Mas não acabamos.
Precisamos descobrir o volume
da parte interna da figura
que será retirada.
Então vamos subtrair esse volume.
Nossos valores de x, de novo,
estão entre zero e um.
Vamos pensar sobre esses discos.
Vamos pensar-- vamos construir um disco
na parte de dentro, bem aqui.
Se eu construir um disco interno.
Agora estou cavando a parte cônica disso.

Thai: 
คุณอาจบอกว่า โอเค 
x กำลังสองลบ x เท่ากับ 0
คุณแยก x ออกมาได้
คุณจะได้ x คูณ x ลบ 1 เท่ากับ 0
ไม่ว่าแบบไหน มันเท่ากับ 0 ได้
x เท่ากับ 0 หรือ x ลบ 1 เท่ากับ 0
แล้วคุณจะได้ x เท่ากับ 0 หรือ x เท่ากับ 1
 
ซึ่งให้ขอบของการอินทิเกรต
อันนี้ไปจาก x เท่ากับ 0 ถึง x เท่ากับ 1
สำหรับรูปทรงข้างนอก
เราหาปริมาตรได้แล้ว
แต่เรายังไม่เสร็จ
เรายังต้องหาปริมาตร
ข้างในรูปทรงที่เราจะเอาออกมา
เราจะลบปริมาตรนั้น
เราจะลบปริมาตรนั้น
ค่า x เหมือนเดิม อยู่ระหว่าง 0 กับ 1
ลองคิดถึงจานเหล่านี้กัน
ลองคิดดู ลองสร้างจาน
ข้างใน ตรงนี้
ถ้าผมสร้างจานข้างใน
ตอนนี้ผมจะตัดส่วนที่เป็นกรวยออก

English: 
You could say OK, x squared
minus x is equal to 0.
You could factor out an x.
You get x times x
minus 1 is equal to 0.
And so either one of
these could be equal to 0,
so x is equal to 0 or x
minus 1 is equal to 0.
And then you get x equals
0 or x is equal to 1.
Which gives us our
boundaries of integration.
So this goes from x
equals 0 to x equals 1.
And so for the
outside of our shape
we can now figure
out the volume.
But we're not done.
We also need to
figure out the volume
of the inside of our shape
that we're going to take out.
So we're going to
subtract out that volume.
So we're going to
subtract out of volume.
Our x values, once again,
are going between 0 and 1.
And so let's think
about those disks.
So let's think about,
let's construct a disk
on the inside, right over here.
So if I construct a
disk on the inside.
So now I'm carving out
the cone part of it.

Korean: 
x²-x=0이니까
x로 묶어보면
x(x-1)=0의 식을 얻을 수 있습니다
x나 x-1이 0이 되어야 하므로
x=0 또는 x-1=0이어야 합니다
따라서 x=0 또는 x=1의 결과를 얻을 수 있습니다
따라서 x=0 또는 x=1의 결과를 얻을 수 있습니다
이것이 적분의 범위입니다
x=0부터 1까지 적분하면 됩니다
바깥쪽 물체의
부피를 알 수 있습니다
그러나 아직 끝나지 않았습니다
바깥쪽 모양에서 제외할
안쪽 모양의 부피를 알아내어야 합니다
안쪽 모양의 부피를 빼면 됩니다
구한 부피를 제외하면 됩니다
x값은 또다시 0과 1 사이가 되겠습니다
원판에 대해 다시 생각해봅시다
원판을 여기 안쪽에
그려봅시다
안쪽에 원판을 그릴 수 있습니다
원뿔 모양 부분을 파내고 있습니다

Bulgarian: 
Може да кажеш, че
х^2 е равно на 0.
Изнасяш х пред скоби.
Получава се х(х – 1) = 0.
Значи един от двата
множителя трябва да е 0,
така че или х = 0, или (х – 1) = 0.
И получаваме х = 0 или х = 1.
Което ни дава границите 
за интегриране.
Това е в интервала от х = 0 до х = 1.
Сега можем да намерим
обема на външното тяло.
Но не сме готови.
Трябва да намерим и обема
на вътрешното тяло, което
ще извадим.
Ще извадим това тяло.
Ще извадим този обем.
Стойностите на х отново
са между 0 и 1.
Да помислим за тези дискове.
Например, да направим диск,
който е вътрешен, например тук.
Ще направя един вътрешен диск.
Сега издълбаваме
сърцевината.

Thai: 
พื้นที่หน้าของจานหนึ่งเป็นเท่าใด?
มันจะเท่ากบพายคูณรัศมีกำลังสอง
ในกรณีนี้ รัศมีจะ
เท่ากับค่าของฟังก์ชันข้างใน
ซึ่งก็คือแค่ x
อันนี้ก็แค่ y เท่ากับ x
แล้วเราจะคูณมันด้วยความลึก
คูความลึกของจานแต่ละแผ่น
แต่ละแผ่นจะหนา dx
ถ้าคุณนึกถึงเหรียญที่บางเฉียบ
ตรงนี้
มันจะเท่ากับ dx
แล้วปริมาตร ขนมทรัฟเฟิลที่คว้านรูปกรวย
ออก จะเท่ากับอินทิกรัลนี้
ลบอินทิกรัลนี่ตรงนี้
และเราหาค่ามันแบบนั้นได้
หรือเราบอกได้ว่า โอเค เราแยก
พายออกทั้งสองข้างได้
มันมี มันมีวิธี
เขียนได้หลายวิธี
แต่ลองหาค่ามันอย่างนี้
แล้วเราจะเขียนโดยทั่วไปในวิดีโอหน้า
อันนี้จะเท่ากับอินทิกรัลจำกัดเขตจาก 0
ถึง 1
คุณเอาพายออกมา
รากที่สองของ x กำลังสองจะ

Korean: 
원판의 면의 넓이는 무엇인가요?
π 곱하기 반지름의 제곱일 것입니다
이 경우, 반지름은
안쪽 모양의 식과 같을 것이고
안쪽 모양의 식은 y=x입니다
안쪽 모양의 식은 y=x입니다
구한 면의 넓이를 두께와 곱해주어야 합니다
각 원판의 두께와 말이지요
모든 원판은 dx의 두께를 가질 것입니다
무한히 얇은 두께를 가지는
동전을 생각해보십시오
두께는 dx입니다
따라서, 원뿔 모양으로 파내어진 트뤼플 형태의 부피는
이 정적분식에서
이 정적분식을 뺀 값입니다
쉽게 계산할 수 있습니다
일단 두 식에서 모두
π를 앞으로 빼낼 수 있을 것입니다
이 식은 다양한 방법으로
쓸 수 있는데요
그냥 이렇게 계산해 봅시다
다음 영상에서 일반화하겠습니다
이 식은 0부터 1까지의 정적분식과
같습니다
π를 밖으로 빼내고
(√x)²을 계산하면

Bulgarian: 
Каква е площта на повърхността
на тези дискове?
Тя ще бъде π по квадрата
на радиуса.
В този случай радиусът
е равен на стойността
на вътрешната функция,
която е просто х.
Това е равно просто на х.
После го умножаваме 
по дебелината,
по дебелината на всеки
от тези дискове.
Всеки диск ще има дебелина dх.
Представи си монета,
която е безкрайно тънка, ето тук.
Това ще е dх.
Значи обемът на нашия
трюфел с издълбан конус вътре
ще бъде равен на този интеграл
минус този интеграл ето тук.
Можем да го сметнем 
по следния начин.
Можем даже да кажем, че
можем да изнесем отпред
пи и от двата интеграла.
Всъщност има много начини
да запишем това.
Но нека да го сметнем така,
а после ще обобщим 
в следващото видео.
Значи това ще е равно на 
определен интеграл от 0 до 1.
Изнасяме пи отпред.

Portuguese: 
Qual a área da face de um desses discos?
Bem, será pi vezes o raio ao quadrado.
Nesse caso, o raio será igual
ao valor dessa função interna,
que é apenas x.
E isso é apenas y é igual a x.
E então vamos multiplicar
pela profundidade.
Vezes a profundidade de cada disco.
E cada um desses discos terá
profundidade dx.
Se imaginarmos uma moeda que tenha
profundidade infinitamente fina bem aqui.
Então será dx.
E então o volume da nossa trufa
com um cone escavado,
será essa integral menos
essa integral aqui.
E podemos avaliar desse jeito.
Ou poderíamos dizer, OK, podemos
fatorar o pi de ambos.
Existem, na verdade, várias
formas de escrever.
Mas vamos apenas avaliar assim, e então
vou generalizar no próximo vídeo.
E isso será igual a integral definida
de zero até um.
Você tira o pi para fora da integral.

English: 
What is the area of the
face of one of those disks?
Well it's going to be pi
times the radius squared.
In this case, the
radius is going
to be equal to the value of
this inner function, which
is just x.
And so this is just
y is equal to x.
And then we're going to
multiply it times the depth,
times the depth of
each of these disks.
And each of these disks are
going to have a depth of dx.
If you imagine a quarter that
has an infinitely-thin depth
right over here.
So it's going to be dx.
And so the volume our, kind of
our truffle with a cone carved
out, is going to
be this integral
minus this integral
right over here.
And we could evaluate
it just like that.
Or we could even say,
OK we could factor out
a pi out of both of them.
There are actually,
there's multiple ways
that we could write it.
But let's just
evaluate it like this,
and then I'll generalize
it in the next video.
So this is going to be equal
to the definite integral from 0
to 1.
You take the pi outside.
Square root of x
squared is going

Portuguese: 
Raiz quadrada de x ao quadrado
será xdx menos a integral,
podemos fatorar o pi.
De zero até um de x ao quadrado dx.
E podemos dizer que será igual a
pi vezes a antiderivada de x,
que é apenas x ao quadrado
sobre dois avaliado de zero até um,
menos pi, vezes a antiderivada
de x ao quadrado,
que é x ao cubo sobre três
avaliado de zero até um.
Essa expressão é igual a--
vou mudar as cores porque
o verde está ficando monótono--
pi vezes um ao quadrado sobre dois
menos zero ao quadrado
sobre dois.
Poderia escrever ao quadrado
Um ao quadrado sobre dois menos zero
ao quadrado sobre dois, menos pi vezes

Bulgarian: 
Квадратен корен от х квадрат
е просто х, dх,
минус интеграл, отново
изнасяме пи отпред,
от 0 до 1 от х^2, dх.
И можем да кажем, че това
е равно на пи по
примитивната функция на х,
което е просто х^2 върху 2,
изчислено в интервала от 0 до 1,
минус пи по примитивната
функция на х^2,
която е х^3 върху 3,
изчислена от 0 до 1.
Този израз е равен на...
ще сменя произволно цвета,
защото това зелено става
много еднообразно –
пи по (1^2)/2 минус (0^2)/2.
Това е на квадрат.
(1^2)/2 минус (0^2)/2,
минус пи по

Thai: 
เท่ากับ x dx ลบอินทิกรัล เราแยกพายออกมา
จาก 0 ถึง 1 ของ x กำลังสอง dx
และเราบอกได้ว่า อันนี้จะ
เท่ากับพายคูณปฏิยานุพันธ์ของ x
ซึ่งก็คือ x กำลังสองส่วน 2 หาค่าจาก 0 ถึง 1
ลบพาย คูณปฏิยานุพันธ์ของ x กำลังสอง
ซึ่งก็คือ x กำลังสามส่วน 3 หาค่าจาก 0 ถึง 1
พจน์นี้เท่ากับ -- ผม
จะเปลี่ยนสีตามใจ
เพราะสีเขียวมันซ้ำซากแล้ว -- 
พายคูณ 1 กำลังสอง
ส่วน 2 ลบ 0 กำลังสองส่วน 2
ผมเขียนกำลังสองได้
1 กำลังสองส่วน 2 ลบ 0 กำลังสองส่วน 2
ลบพายคูณ 1

English: 
to be x dx minus the integral,
we can factor the pi out.
From 0 to 1 of x squared dx.
And we could say
this is going to be
equal to pi times the
antiderivative of x, which
is just x squared over
2 evaluated from 0 to 1,
minus pi, times the
antiderivative of x squared,
which is x to the third over
3 evaluated from 0 to 1.
This expression is
equal to-- and I'm
going to arbitrarily
switch colors
just because the green's getting
monotonous-- pi times 1 squared
over 2 minus 0 squared over 2.
I could write squared.
1 squared over 2 minus 0
squared over 2, minus pi times 1

Korean: 
x dx 빼기 정적분식이 될 것이고
여기서도 π를 빼낼 수 있습니다
0부터 1까지 x²을 적분해주면 됩니다
π 곱하기
x의 역도함수는 x²이고
이를 0부터 1까지 적분해준 값에서
π 곱하기
x²의 도함수인 x³/3을 0부터 1까지 적분한 값을 빼준 것과 같습니다
이 식은
색을 바꾸어 보겠습니다
계속 초록색으로만 써서
π 곱하기 1/2 - 0
제곱이라고 쓰겠습니다
1의 제곱 나누기 2 빼기 0의 제곱 나누기 2

Korean: 
빼기 π 곱하기 1 세제곱 나누기 3 빼기 0 세제곱 나누기 3
이를 계산해 보면
파란색으로 해보겠습니다
단순하네요
이건 바로 0이고
1의 제곱 나누기 2는 그냥 1/2이고
따라서 π/2네요
옆의 식은 -π/3의 결과가 나옵니다
이를 단순화시키는 과정은
분수 계산밖에 없는데요
공통분모를 찾아봅시다
공통분모를 6이라고 하면
3π/6이고
3π/6 - 2π/6가 됩니다
3π/6 - 2π/6가 됩니다
3π/6 - 2π/6가 됩니다
마지막으로 3 곱하기
빼기 2 곱하기
결국 π/6가 나옵니다

Portuguese: 
um ao cubo sobre três menos
zero ao cubo sobre três.
E então temos, isso igual a--
vou fazer no mesmo azul--
isso é isso simplificado.
Isso aqui é apenas zero.
Isso é um ao quadrado sobre dois,
que é meio.
Isso é apenas pi sobre dois,
meio vezes pi menos--
bem, isso é zero, isso é um terço,
menos pi sobre três.
E para simplificar, é somente
subtrair frações.
E então achamos um
denominador comum.
Denominador comum é seis.
Isso será três pi sobre seis.
Isso é três pi sobre seis
menos dois pi sobre seis.
pi sobre três é dois pi sobre seis,
pi sobre dois é três pi sobre seis.
E terminamos com três de alguma coisa
menos dois de alguma coisa,
e então um de alguma coisa.
Terminamos com um pi sobre seis.

English: 
to the third over 3 minus
0 to the third over 3.
And so we get, this
is equal to-- let
me do it in that
same blue color--
so this is this simplified.
This is just 0 right over here.
This is 1 squared over
2, which is just 1/2.
So it's just pi over 2, 1/2
times pi minus-- well this
is just 0, this is
1/3, minus pi over 3.
And then to simplify
this, it's just
really subtracting fractions.
So we can find a
common denominator.
Common denominator is 6.
This is going to be 3 pi over 6.
This is 3 pi over 6
minus 2 pi over 6.
pi over 3 is 2 pi over 6,
pi over 2 is 3 pi over 6.
And we end up with, we
end up with 3 of something
minus 2 of something, you
end up with 1 of something.
We end up with 1 pi over 6.

Thai: 
กำลังสามส่วน 3 ลบ 0 กำลังสามส่วน 3
แล้วเราได้ อันนี้เท่ากับ --
ขอผมใช้สีฟ้าสีเดิมนะ --
อันนี้คืออันนี้ลดรูปลง
นี่ก็แค่ 0 ตรงนี้
นี่คือ 1 กำลังสองส่วน 2 ซึ่งก็คือแค่ 1/2
นี่ก็แค่พายส่วน 2, 1/2 คูณพายลบ -- นี่ก็
แค่ 0 นี่คือ 1/3, ลบพายส่วน 3
แล้วเวลาจัดรูป มันก็แค่
การลบเศษส่วน
เราก็หาตัวส่วนร่วมได้
ตัวส่วนร่วมคือ 6
มันจะเท่ากับ 3 พายส่วน 6
นี่คือ 3 พายส่วน 6 ลบ 2 พายส่วน 6
 
พายส่วน 3 คือ 2 พายส่วน 6
พายส่วน 2 คือ 3 พายส่วน 6
แล้วเราได้ เราได้ 3 ของอะไรสักอย่าง
ลบ 2 ของอะไรสักอย่าง คุณจะได้ 1 ของตัวนั้น
เราจึงได้ 1 พายส่วน 6

Bulgarian: 
(1^3)/3 минус (0^3)/3.
И така получаваме, 
това е равно на...
ще използвам същия син цвят –
това е това, опростено.
Това тук е просто нула.
Това е 1 на квадрат върху 2,
което е просто 1/2.
Значи това е π/2 минус...
това е нула, това е 1/3,
минус π/3.
За да опростим това,
трябва просто да извадим дробите.
Да намерим общ знаменател.
Общият знаменател е 6.
Това е 3π/6.
3π/6 минус 2π/6.
π/3 е 2π/6, π/2 е равно на 3π/6.
Накрая получаваме 3 по нещо минус 
2 по нещо, което е 1 от това нещо.
Получаваме 1π/6.

Portuguese: 
E acabamos.
Fomos capazes de achar o volume
dessa estranha trufa.
[Legendado por Miguel Infante]

English: 
And we are done.
We were able to find the
volume of that wacky kind
of gutted-out truffle.

Bulgarian: 
И сме готови.
Намерихме обема
на това странно тяло,
което прилича на издълбан
трюфел.

Thai: 
เราก็เสร็จแล้ว
เราหาปริมาตรของรูปทรัฟเฟิลคว้านไส้
ประหลาดนั้นได้แล้ว
 

Korean: 
끝났습니다
안쪽이 파인 트뤼플 모양의
부피를 구했습니다
커넥트 번역 봉사단 | 서윤아
