
Korean: 
이번에 초점을 맞출 영역은
이번에 초점을 맞출 영역은
여기 노란색으로 그려진
x=(y-1)²의 그래프와
파란색의 y=x-1의 그래프 사이의
영역입니다
이 영역을 아래 점선으로 표시된
y=-2를 축으로 하여
회전시켜 새로운 도형을 만들겠습니다
이 도형의 부피를 구해볼까요?
이 부피를 계산하기 위해
디스크 방법과 원주각 방법을
모두 사용할 수 있습니다
디스크 방법으로는
이렇게 생긴 원판을 만들면 되겠죠
이를 x에 대한 적분으로 계산하면 됩니다
이렇게 하면
적분을 두 번 해야 합니다
왜냐하면 영역의 아랫부분이
x값에 따라 바뀌기 때문입니다
노란색의 함수를 두 개로 나눠서
위쪽과 아래쪽을 표현하는 각각으로
나누어야 합니다
그렇게 계산하면 답을 구할 수 있죠

Thai: 
 
ผมจะวาดเขตระหว่างเส้นโค้งสองเส้นตรงนี้
ระหว่างเส้นโค้งสีเหลือง -- นิยามฟังก์ชันของ y
ว่า x เท่ากับ y ลบ 1 กำลังสอง -- และ
เส้นสีเขียว
ฟ้านี่ -- y เท่ากับ x ลบ 1
ผมจะนำเขตนี่ตรงนี้มา
และผมจะหมุนมันรอบเส้นตรง
y เท่ากับลบ 2 ได้รูปนี้ที่
เป็นเหมือนด้านหน้าเครื่องยนต์เจ็ต 
อะไรพวกนั้น
และเราอยากหาว่าปริมาตรของมันเป็นเท่าใด
คุณหาคำตอบได้
ด้วยวิธีแบบจานหรือแบบเปลือก
ในวิธีแบบจาน คุณจะสร้าง
จานที่เป็นแบบนี้
และคุณจะอินทิเกรตเทียบกับ x
คุณต้องแบ่งปัญหานี้ดีๆ
เพราะคุณมีขอบล่างต่างกัน
คุณต้องแบ่งรูปนี้เป็นฟังก์ชันสองตัว
ฟังก์ชันบนกับล่าง
สำหรับช่วง x นี้
แล้วก็อีกตัวสำหรับช่วงของ x
แต่คุณใช้วิธีแบบจานได้

English: 
I'm going to take the region
in between the two curves
here, between the yellow curve--
defined as a function of y
as x is equal to y minus 1
squared-- and this bluish-green
looking line-- where y
is equal to x minus 1.
So I'm going to take this
region right over here,
and I'm going to rotate
it around the line,
y equals negative 2,
to get this shape that
looks like the front of a jet
engine or something like that.
And we're going to want to
figure out what its volume is.
And you can actually
approach this
with either the disk
method or the shell method.
In the disk method,
you would create
disks that look like this.
And you would be doing
integrating with respect to x.
You will have to break up
the problem appropriately,
because you have a
different lower boundary.
You would have to break
this up into two functions,
an upper function
and a lower boundary
for this interval in x.
And then a different one
for that interval in x.
But you could use
the disk method.

Portuguese: 
Tomaremos a região entre duas curvas,
uma definida como função de y,
onde x igual a y menos um ao quadrado,
e outra onde y é igual a x menos um.
Pegarei esta região aqui,
e vou girar ao redor da linha
y igual dois negativo para obter uma forma
que parece com a frente de um avião.
Vamos descobrir seu volume.
Para isso podemos usar o
método do disco ou das cascas cilíndricas.
No método do disco, criaríamos
discos como este,
integrando em relação a x.
Mas temos que dividir o problema 
corretamente
porque temos bordas inferiores distintas.
Precisamos usar duas funções,
uma função superior e outra para a borda
inferior, no intervalo em x,
E a outra para o outro intervalo em x.
Mas podemos usar método do disco.

Bulgarian: 
Искам да взема участъка
между графиките на двете функции,
жълтата крива, дефинирана 
от функцията
х = (у – 1)^2
и тази синкавозелена крива,
съответстваща на у = х – 1.
Значи ще взема този участък 
ето тук,
и ще го завъртя около 
правата у = –2,
при което се получава
ротационно тяло,
което изглежда като челото
на самолетен двигател.
Искаме да намерим
обема на това тяло.
Можеш да използваш за
тази задача
и метода на дисковете, и
метода на черупката.
При метода на дисковете
си представяш
дискове, които изглеждат
ето така,
и интегрираш спрямо х.
Ще разделиш задачата
по съответния начин,
защото имаш различна
долна граница.
Трябва да я разделиш на две
функции,
горна функция и долна
граница за този интервал за х.
И после друга функция за 
този интервал за х.
Но можеш да използваш
метода на дисковете.

English: 
But instead of
that, we don't feel
like breaking up the functions
and doing all of that.
So we're going to do
it in the shell method,
especially because we've already
expressed one of our functions
as a function of y.
And this one won't
be too hard to do.
So what we're going
to do, once again,
is imagine constructing
these little rectangles that
have height dy.
And what we're going to do
is rotate those rectangles
around the line y
equals negative 2.
So let me draw that same
rectangle over here.
And when you do that,
you construct a shell.
So let me do that.
So, go between these two points.
And then this is what it would
look like when it's down here.
And then, let me make it
clear that this constructs
a shell of thickness
dy or of depth dy.
So let me do it like that.
So that's my shell, and it
has some thickness to it.
And that thickness is dy.

Portuguese: 
Mas, ao contrário, não queremos
quebrar as funções e tudo mais.
Então vamos usar o método 
das cascas cilíndricas
Porque já expressamos uma das funções como
uma função de y.
Isso não será muito difícil de fazer.
O que vamos fazer, de novo,
é imaginar a construção de pequenos
retângulos com altura dy.
Vamos girar os retângulos
em torno de y igual a dois negativo.
Vamos desenhar o mesmo retângulo aqui.
Quando fazemos isso construímos a casca.
Deixe-me fazer isso.
Vamos entre os dois pontos.
E assim é quando descemos aqui.
E assim, deixamos claro que constrói uma
casca de espessura dy ou profundidade dy
Deixe-me fazer assim.
Essa é a casca e tem uma espessura,
que é dy.

Thai: 
แทนที่จะทำอย่างนั้น เราไม่
อยากแบ่งฟังก์ชันและทำอะไรพวกนั้น
เราจะทำด้วยวิธีแบบเปลือก
ยิ่งถ้าเรามีฟังก์ชันหนึ่ง
อยู่ในรูปของ y แล้ว
และอันนี้ทำได้ไม่ยากนัก
สิ่งที่เราจะทำ เหมือนเดิม
คือจินตนาการการสร้างสี่เหลี่ยมมุมฉากเล็กๆ
ที่สูง dy
และสิ่งที่เราจะทำ คือหมุนสี่เหลี่ยมเหล่านี้
รอบเส้นตรง y เท่ากับลบ 2
ขอผมวาดสี่เหลี่ยมมุมฉากเดียวกันตรงนี้นะ
เมื่อคุณทำอย่างนั้น คุณจะสร้างเปลือกขึ้นมา
ขอผมทำนะ
ไประหว่างสองจุดนี้
นี่คือหน้าตาของมันเมื่อมันลงมาตรงนี้
แล้ว ขอผมบอกให้ชัดว่า อันนี้สร้าง
เปลือกหนา dy หรือลึก dy
ขอผมทำอย่างนั้นนะ
นั่นคือเปลือกของผม และมันมีความหนาอยู่
ความหนานั้นคือ dy

Bulgarian: 
Но вместо това не бихме искали
да разделяме функциите и всичко това.
Ще използваме метода на
черупката,
по-специално защото вече
едната функция е изразена чрез у.
И това няма да е толкова
трудно.
Затова отново ще си представим,
че построяваме тези малки
правоъгълници,
които имат височина dy.
Ще завъртим тези 
правоъгълници
около оста у = –2.
Ще направя същия правоъгълник
ето тук.
Когато направиш това, 
получаваш черупката.
Ще направя това между
ето тези две точки.
Ето така ще изглежда
ето тук долу.
Искам да поясня, че получаваме
черупка с дебелина dy 
или дълбочина dy.
Ще го направя ето така.
Това е моята черупка, която
има някаква дебелина.
Тази дебелина е dy.

Korean: 
만약 함수를 분리하는
복잡한 작업을 하고 싶지 않다면
원주각 방법을 사용해도 됩니다
특히 이미 함수를 y에 대해
표현하였기 때문에
더 사용하기 쉽겠죠
이렇게 하면 별로 복잡하지 않습니다
우선 이 영역을
높이 dy의 작은 직사각형으로
나눠보겠습니다
이 작은 직사각형은
y=-2를 축으로 회전합니다
같은 직사각형을
여기도 그려볼게요
이 직사각형으로 원기둥 껍질을 만듭니다
한번 해 볼까요?
이 두 점 사이의 부분이
회전해 내려오면 여기에 오겠네요
지금 그리는 껍질은
두께가 dy입니다
이를 고려해서 그리겠습니다
원기둥이 그려졌습니다
이 원기둥의 두께는 dy입니다

Bulgarian: 
Това е дебелината на 
черупката.
Ще я щриховам малко.
Искам да съм сигурен, че
виждаш тримерно черупката.
И както във всички
подобни задачи,
целта ни е просто да намерим
обема на всяка черупка.
После ще ги заместим за 
определено у в нашия интервал.
И после интегрираме за всяко у
в интервала.
Правихме го вече
много пъти.
Най-напред да определим
колко е радиусът на тези черупки.
Тук в цикламено означавам –
колко е радиусът на нещо подобно?
Това е равно на
разстоянието между у = –2
и нашето у в тази точка.
Значи това разстояние 
ето тук е равно на у.
И после имаме още 2.

English: 
So that's the
thickness of my shell.
And let me shade
it in a little bit.
Make sure you can see the
3-dimensionality of my shell.
So like we've done with
all of these problems,
our goal is to really
just figure out
what the volume
of each shell is.
And then we can enter it for
a specific y in our interval.
And then we integrate along
all the y's of our interval.
And we've done this
many times before.
The first thing
we think about is
what's the radius
one of these shells?
So what I'm doing
right here in magenta,
what is the radius of
something like that?
Well, it's essentially
going to be
the distance between y
is equal to negative 2
and our y value for
that specific y.
So this distance
right over here is y.
And then we're going
to have another 2.

Thai: 
นั่นคือความหนาของเปลือกผม
ขอผมแรเงาหน่อย
ดูให้แน่ใจว่าเปลือกผมมีความเป็นสามมิติ
อย่างที่เราได้ทำกับปัญหาเหล่านี้ทั้งหลาย
เป้าหมายของเราคือหา
ว่าปริมาตรของเปลือกแต่ละอันเป็นเท่าใด
แล้วเราก็ใส่ y เฉพาะในช่วงของเรา
แล้วเราก็อินทิเกรตตาม y ของช่วงเรา
และเราทำอันนี้มาหลายครั้งแล้ว
อย่างแรกที่เราคิดคือว่า
รัศมีของเปลือกเหล่านี้เป็นเท่าใด?
สิ่งที่ผมจะทำตอนนี้ด้วยสีบานเย็น
รัศมีของรูปแบบนี้จะเป็นเท่าใด?
มันจะเท่ากับ
ระยะระหว่าง y เท่ากับลบ 2
กับค่า y สำหรับค่า y เฉพาะนั้น
 
ระยะนี่ตรงนี้คือ y
แล้วเราจะมีอีก 2

Portuguese: 
Essa é a espessura da concha.
Vou colorir aqui um pouco.
Veja o 3D da minha casca.
Quando terminamos estes problemas,
o objetivo é determinar
o volume de cada casca.
Assim podemos entrar com um intervalo y.
E integramos em todo o intervalo de y.
Já fizemos isso várias vezes,
primeiro pensamos em qual
o raio de uma das cascas?
O raio do desenho em magenta
essencialmente será a distância
entre y igual dois negativo e o
um valor específico de y.
A distância aqui é y.
E temos outro 2.

Korean: 
명암을 표시해서
조금 더 잘 그려보겠습니다
이 그림은 3차원을 그린 것임을 잊지 마세요
이전 영상들에서도 해보았듯이
우리가 해야 할 것은
이 원기둥의 부피를 구하는 것입니다
그런 후에 y의 범위를 구해서
그 범위에서 적분하면 됩니다
이미 여러 번 해 본 작업이지요?
첫 번째로 생각해볼 것은
이 껍질의 반지름입니다
이 분홍색 부분의 반지름은
얼마일까요?
이 값은 y부터
y=-2까지의 거리와
같습니다
같습니다
그러므로 이 길이가 y이고
2를 더해주면

Portuguese: 
A distância será y mais dois.
Outra forma de pensar é que isso
é igual a y menos dois negativo para
saber a distância, que será y mais dois.
Então o raio de uma das cascas
será y mais dois.
Se o raio é y mais dois, então sabemos que
o comprimento da circunferência será
dois pi vezes y mais dois.
Então a área da superfície externa
da casca, destacadas aqui,
será o comprimento da circunferência
vezes a largura desta concha vezes
esta distância aqui.
Ou podemos dizer vezes esta distância.
E qual é a distância?
Lembre-se: Queremos tudo como função de y.
Isto será função superior como função de y
menos a função inferior.
Pensamos sobre a função superior,

English: 
So the whole distance
is going to be y plus 2.
Another way of thinking
about it is this
is essentially y
minus negative 2
to get the distance, which
is going to be y plus 2.
So the radius of one
of our little shells
is going to be y plus 2.
If the radius is
y plus 2, then we
know that the circumference
of this circle right over here
is going to be 2
pi times y plus 2.
And then the surface area,
the outside surface area,
of the shell, the
stuff out here,
is just going to be that
circumference times I guess
you could say the width
of this shell times
this distance right over here.
Or we could say times this
distance right over here.
And what is that distance?
Remember, we want everything
expressed as a function of y.
Well, it's going to be the upper
function as a function of y
minus the lower function.
And we think about
the upper function,

Thai: 
ระยะทั้งหมดจึงเท่ากับ y บวก 2
วิธีคิดอีกอย่างคือว่า
นี่ก็คือ y ลบลบ 2
เพื่อให้ได้ระยะนี้ จะเท่ากับ y บวก 2
รัศมีของเปลือกเล็กๆ อันหนึ่ง
จะเท่ากับ y บวก 2
ถ้ารัศมีคือ y บวก 2 เราจะ
รู้เส้นรอบวงของวงกลมนี่ตรงนี้
จะเท่ากับ 2 พายคูณ y บวก 2
แล้วพื้นที่ผิว พื้นที่ผิวด้านนอก
ของเปลือก เปลือกข้างนอกนี้
จะเท่ากับเส้นรอบวงนั่นคูณ
จะเรียกว่าความกว้างของเปลือกนี้คูณ
ระยะนี่ตรงนี้ก็ได้
หรือเราบอกได้ว่า คูณระยะนี่ตรงนี้
แล้วระยะนั้นเป้นเท่าใด?
นึกดู เรามีทุกอย่างเขียนในรูปของ y
มันจะเป็นฟังก์ชันบน เป็นฟังก์ชันของ y
ลบฟังก์ชันล่าง
และเราคิดถึงฟังก์ชันบน

Bulgarian: 
Цялото това разстояние е у + 2.
Друг начин да разсъждаваме
за това е,
че всъщност ще извадим –2 от у,
за да получим разстоянието, което
ще бъде у + 2.
Радиусът на една от тези
малки черупки
ще бъде у + 2.
Ако радиусът е у + 2,
тогава можем да намерим
обиколката на тази окръжност,
която ще е 2π по (у + 2).
После лицето на повърхнината –
лицето на околната повърхнина
на тази черупка ето тук,
ще бъде равно на тази 
обиколка по –
по широчината на тази
черупка, по
това разстояние ето тук.
Можем да кажем по това
разстояние ето тук.
И какво е това разстояние?
Искаме всичко да е изразено
като функция от у.
Горната функция на у,
минус долната функция.
Когато определяме горната
функция,

Korean: 
총 길이가 y+2가 됩니다
다른 방법으로 생각해볼까요?
y에서 -2까지의 거리를 구하기 위해
y에서 -2를 빼면 y+2가 됩니다
그러므로 껍질의 반지름은
y+2가 됩니다
반지름이 y+2이므로
분홍색으로 표시된 원주는
2π(y+2)가 됩니다
이번엔 껍질의 겉넓이를 구해봅시다
이 부분이 되겠네요
방금 구한 원주에
이 껍질의 폭을 곱하면 됩니다
이 길이가 되겠죠
왼쪽 그림에서는 이 부분이 되겠네요
이 길이는 얼마일까요?
y에 대해서 길이를
표현해야 합니다
y에 대해서 위쪽 함수에서
아래쪽 함수를 빼면 되겠네요
위쪽 함수는

English: 
it's the function
that's giving us
higher x values
in that interval.
And so this blue function
is the upper function
when we think in terms of y.
But we have to express
it as a function of y.
So let me do that.
So we can rewrite this
as add 1 to both sides,
you get x is equal to y plus 1.
So that is our upper function.
And then this is
our lower function.
If you were to tilt
your head to the right
and look at it that
way, you'll see
that this will be
the upper function,
and this will be
the lower function
for the same value of y.
This gives a higher value
of x than this one does
for the same value of y.
It gives the upper x values.
So, the area is going
to be the circumference
times this dimension.
Let me write this.
The area of one of those shells
is going to be 2 pi times y
plus 2 times the distance
between the upper function.

Bulgarian: 
тя трябва да има по-големи 
стойности за х в този интервал.
Значи тази синя функция
е горната функция,
когато разсъждаваме
по отношение на у.
Но трябва да я изразим
спрямо у.
Ще го направя.
Можем да преработим това, като добавим 
1 към двете страни на равенството,
и получаваме х = у + 1.
Това е горната ни функция.
Това е долната ни функция.
Ако си наклониш главата
надясно и погледнеш по този начин,
ще видиш, че това е
горната функция,
а това е долната функция
за една и съща стойност на у.
Тази ни дава по-висока стойност
за х, отколкото тази,
за една и съща стойност на у.
Тук имаме по-големи
стойности на х.
Значи площта ще бъде
обиколката по този размер.
Ще го напиша.
Площта на една от тези черупки
ще е равна на 2π по (у + 2)
по разстоянието между двете
функции.

Portuguese: 
é a função que nos dá valores máximos
de x neste intervalo.
Esta função azul é a função superior
quando pensamos em termos de y.
Tem que ser expressa como função de y.
Deixe-me fazer isso.
Podemos reescrever adicionando um aos dois
lados tendo x igual y mais um.
Então essa é nossa função superior.
E a nossa função inferior
Se inclinar a cabeça
para direita e olhar desta forma
verá que essa é a função superior,
e esta será a função inferior
para mesmo valor de y.
Isso dá um valor maior para x do que esse-
para mesmo valor de y.
Aqui temos os valores superiores de x
Então a área será uma circunferência
vezes a dimensão.
Irei escrever.
A área de uma casca será dois pi vezes y
mais dois, vezes a distância 
da função superior,

Thai: 
มันคือฟังก์ชันที่ให้ค่า
x ที่สูงกว่าในช่วงนั้น
และฟังก์ชันสีฟ้านี้ คือฟังก์ชันบน
เวลาเราคิดในรูปของ y
แต่เราต้องเขียนมันเป็นฟังก์ชันของ y
ขอผมทำนะ
เราเขียนอันนี้ใหม่ได้โดยบวก 1 ทั้งสองข้าง
คุณจะได้ x เท่ากับ y บวก 1
นั่นคือฟังก์ชันบนของเรา
แล้วอันนี้คือฟังก์ชันล่าง
ถ้าคุณเอียงหัวไปทางขวา
แล้วดูแบบนั้น คุณจะเห็น
ว่าอันนี้จะเป็นฟังก์ชันบน
และนี่จะเป็นฟังก์ชันล่าง
สำหรับค่า y เดียวกัน
อันนี้จะให้ค่า x สูงกว่าอันนี้
เมื่อเทียบค่า y เดียวกัน
มันให้ค่า x ข้างบน
พื้นที่จะเท่ากับเส้นรอบวง
คูณขนาดอันนี้
 
ขอผมเขียนนะ
พื้นที่ของเปลือกอันหนึ่งจะเท่ากับ 2 พายคูณ y
บวก 2 คูณระยะระหว่างฟังก์ชันบน

Korean: 
같은 y의 값에서
더 높은 x값을 가집니다
그러므로 이 파란 부분이
y에 대해 위쪽 함수가 됩니다
y에 대한 함수로 고쳐볼까요?
y에 대한 함수로 고쳐볼까요?
양변에 1을 더해주기만 하면 됩니다
x=y+1이 되네요
이게 위쪽 함수가 됩니다
아래쪽 함수는
이 노란색의 함수입니다
머리를 오른쪽으로 돌려서 본다면
쉽게 이해할 수 있을 겁니다
파란색이 윗부분이 되고
노란색이 아랫부분이 됩니다
같은 y의 값에서 말이죠
어떤 y의 값에 대해
파란색의 함수가
더 큰 x값을 가지게 됩니다
그러므로 껍질의 겉넓이는
아까 구한 원주에
이 길이를 곱하면 됩니다
이 길이를 곱하면 됩니다
적어보도록 하겠습니다
껍질의 겉넓이는 2π에
y+2를 곱한 후에

Portuguese: 
então a distância entre a função y mais 1,
x igual a y mais 1, e a função inferior,
x igual a y menos um ao quadrado.
Colocarei o parênteses na mesma cor.
E se quisermos o volume da casca,
usamos a superfície externa da casca e
multiplicamos pela profundidade que é dy.
Isso nos dá a integral.
O volume de uma casca é
dois pi, vezes y mais dois, vezes
y mais um menos o quadrado de y menos um.
Multiplicamos pela espessura da casca, dy,
e integramos sobre o intervalo.
E teremos o volume.
Qual intervalo?
Talvez consiga ver isso.
Mas conseguimos resolver isso.
Quando esses dois termos se igualam?
Podemos fazer y mais um

English: 
So the distance between the
upper function y plus 1,
x is equal to y plus 1,
and the lower function, x
is equal to y minus 1 squared.
I'll put the parentheses
in that same color.
And then if we want the
volume of that shell,
so we've got the outside surface
area of the shell right now.
We just multiply it by its
depth, which is just dy.
So that sets up our integral.
The volume of one shell--
I'll do it all in one color
now-- is 2 pi times y plus 2
times y plus 1 minus y minus 1
squared.
Then we multiply that times
the depth of each shell, dy.
And then we integrate
over the interval.
So the volume is
going to be this.
And what's the interval?
You might be able to eyeball it.
But we can actually
solve that explicitly.
When do these two
things equal each other?
Well, you could
just set y plus 1

Bulgarian: 
Разстоянието между горната
функция х = (у + 1)
и долната функция х = (у – 1)^2.
Ще сложа скоби в същия цвят.
И за да намерим обема
на черупката,
сега имаме външната повърхнина
на тази черупка.
Просто умножаваме по дебелината,
която е dу.
И сега да конструираме
нашия интеграл.
Обемът на черупката –
ще го направя в нов цвят –
е 2π по (у + 2) по, скоба (у + 1) 
минус (у – 1)^2.
После умножаваме по дебелината
на черупката, т.е. по dy.
И интегрираме в 
някакъв интервал.
Това е обемът, а кой е интервалът?
Може би можеш да
определиш на око.
Но можем да го определим
алгебрично.
Когато тези двете са равни?
Можем да приравним (у + 1)

Thai: 
ระยะระหว่างฟังก์ชันบน y บวก 1
x เท่ากับ y บวก 1 และฟังก์ชันล่าง
x เท่ากับ y ลบ 1 กำลังสอง
 
ผมจะใส่วงเล็บด้วยสีเดียวกันนะ
แล้วถ้าเราต้องการปริมาตรของเปลือกนั้น
เราจะได้พื้นที่ผิวข้างนอกของเปลือกตรงนี้
เราแค่คูณมันด้วยความหนา ซึ่งก็คือ dy
มันตั้งอินทิกรัลขึ้นมา
ปริมาตรของเปลือกอันหนึ่ง -- ผมจะใช้สีเดียว
แล้ว -- คือ 2 พายคูณ y บวก 2 
คูณ y บวก 1 ลบ y ลบ 1
กำลังสอง
แล้วเราคูณมันด้วยความลึกของเปลือก dy
แล้วเราอินทิเกรตตอลดช่วงนี้
ปริมาตรจะเท่ากับค่านี้
แล้วช่วงคืออะไร?
คุณอาจกะด้วยสายตาได้
แต่เราแก้มันได้โดยตรง
สองตัวนี้เท่ากันเมื่อใด?
คุณก็ให้ y บวก 1

Korean: 
위쪽 함수는 y+1에서
아래쪽 함수인 (y-1)²을
빼준 식을 곱하면 됩니다
빼준 식을 곱하면 됩니다
괄호를 같은 색으로 칠게요
이 껍질의 부피는
겉넓이에 두께를 곱하면 되므로
dy를 곱하기만 하면 됩니다
적분할 준비가 되었군요
껍질 한 개의 부피는
2π(y+2)(y+1-(y-1)²)에다가
2π(y+2)(y+1-(y-1)²)에다가
두께 dy를 곱하면 됩니다
이를 범위에 맞게
적분해주면 되겠네요
적분 범위는 어디일까요?
암산으로도 구할 수 있겠네요
하지만 정확하게 써보도록 하겠습니다
두 그래프의 교점을 구해야 합니다
y+1의 값과

Korean: 
(y-1)²의 값이 같은 곳을
찾으면 됩니다
교점을 찾기 위해서
식을 전개해보겠습니다
(y-1)²은 y²-2y+1과 같습니다
이제 정리할 수 있겠네요
양변에서 y를 빼고
1도 빼겠습니다
1도 빼겠습니다
그러면 좌변에는 0만 남고
우변에는 y²-3y가 남게 됩니다
우변에는 y²-3y가 남게 됩니다
이를 인수분해해보면
0=y(y-3)이 되므로
이 식이 성립하는 y는
0 또는 3이겠네요
그림에서도 이를 확인할 수 있습니다
y가 0인 지점에서 두 함수가 만나네요
y가 3인 지점에서도 그렇습니다

Bulgarian: 
да е равно на (у – 1)^2,
да го направим.
Ако приравним (у + 1) 
да е равно на това,
това ще е равно на (у – 1)^2.
Нека да го разложа.
Това ще е у^2 – 2у + 1.
Да видим.
Мога да извадя у от двете страни,
мога да извадя и 1 от
двете страни.
Минус у, минус 1.
И ми остава отляво 0.
А отдясно остава y^2 – 3у.
И това е плюс 0.
Значи 0 е равно на у(у – 3).
Нулите тук са, когато тези
са равни ,
когато у = 0 или у = 3.
И виждаме това ето тук.
Когато у = 0 двете графики
се пресичат.
Когато у = 3, двете 
графики се пресичат.

Portuguese: 
igual ao quadrado de y menos um.
Tomando y mais um igual ao
quadrado de y menos um.
Expandindo temos
y ao quadrado menos dois y mais um.
Vejamos
Podemos subtrair y nos dois lados
e subtrair um dos dois lados.
Menos y, menos um.
Então temos zero no lado esquerdo,
e no lado direito, tenho y ao quadrado
menos três y.
E isso será igual a zero.
Então zero é igual a y vezes y menos três,
O produto é zero quando
y é igual a zero ou y igual três.
E vemos isso bem aqui.
Em y igual a zero as funções interceptam.
Em y igual três, as funções interceptam.

English: 
to be equal to y minus 1
squared, so let's do that.
So if we set y plus 1,
this, to be equal to that,
it's going to be equal
to y minus 1 squared.
So let me expand that out.
That's going to be y
squared minus 2y plus 1.
And let's see.
I could subtract
y from both sides
and I could subtract
1 from both sides.
Minus y, minus 1.
And then I am left with,
on the left hand side, 0.
And on the right hand side,
I have y squared minus 3 y.
And that's it, plus 0.
So 0 is equal to
y times y minus 3.
So the zeros of
this, when these are
equal are when y is equal
to 0 or y is equal to 3.
And we see that right over here.
When y is equal to 0, these
two functions intersect.
And when y is equal to 3,
these two functions intersect.

Thai: 
เท่ากับ y ลบ 1 กำลังสองได้ ลองทำดู
ถ้าเราให้ y บวก 1 อันนี้ เท่ากับอันนั้น
มันจะเท่ากับ y ลบ 1 กำลังสอง
ขอผมกระจายมันออกมานะ
มันจะเท่ากับ y กำลังสองลบ 2y บวก 1
ลองดู
ผมลบ y ทั้งสองข้างได้
และผมลบ 1 จากทั้งสองข้างได้
ลบ y, ลบ 1
แล้วผมจะเหลือ ทางซ้ายมือเป็น 0
และทางขวามือ ผมมี y กำลังสองลบ 3y
 
แค่นั้นแหละ บวก 0
0 เท่ากับ y คูณ y ลบ 3
รากของสมการนี้ เมื่อสมการ
เป็นจริง คือเมื่อ y เท่ากับ 0 หรือ y เท่ากับ 3
และเราเห็นมันตรงนี้
เมื่อ y เท่ากับ 0 ฟังก์ชันสองตัวนี้ตัดกัน
และเมื่อ y เท่ากับ 3 ฟังก์ชันสองตัวนี้ตัดกัน

Thai: 
ช่วงของเราจึงไปจาก y เท่ากับ 0
ถึง y เท่ากับ 3
เมื่อใช้วิธีแบบเปลือก เรา
สามารถตั้งอินทิกรัลจำกัดเขตได้
เราคิดได้ว่าเราจะหาค่านี้ได้อย่างไรต่อไป

Portuguese: 
Então o intervalo será de y igual a zero
até y igual a três.
Então, com o método das cascas cilíndricas
escrevemos nossa integral definida.
E podemos então pensar 
em como estimar seu valor.
Legendado por: [Fabiana]
Revisado por: [Tatiana F. D'Addio]

English: 
So our interval is going
to be from y is equal to 0
to y is equal to 3.
So using the shell
method, we have
been able to set up
our definite integral.
And now we can think about how
we can evaluate this thing.

Korean: 
따라서 적분 범위는
y=0부터 y=3까지입니다
원주각 방법을 이용해서
정적분을 구한 것입니다
계산은 생략하도록 하겠습니다
커넥트 번역 봉사단 | 박혜준

Bulgarian: 
Значи интервалът ще бъде
от у = 0 до у = 3.
Значи с помощта на метода
на черупката
успяхме да конструираме
нашия определен интеграл.
И сега можем да видим
как ще сметнем това.
