
English: 
What we're going to do in this
video is take the function y
is equal to the cube
root of x and then rotate
this around the x-axis.
And if we do that, we
get a solid of revolution
that looks like that.
And we're doing it between x is
equal to 0 and x is equal to 8.
And you get something
that looks like this.
And you could find the
volume of this actually
quite easily using
the disk method.
But just to show you that you
could do it an alternate way,
we're going to use
the shell method.
But we're going to use
the shell method now
to rotate around a horizontal
line, and specifically
the x-axis.
So how would we do this?
Well, what we want to do,
what you could imagine,
is constructing a rectangle
that looks like this.
Let me do it in
this salmon color.
So you have a rectangle that
looks something like that.
Its depth or its height,
you could say, is dy.
And then its length
right over here

Thai: 
 
สิ่งที่เราอยากทำในวิดีโอนี้คือนำฟังก์ชัน y
เท่ากับรากที่สามของ x มาหมุน
รอบแกน x
และถ้าเราทำอย่างนั้น เราจะได้ทรงตันจากการหมุน
ที่เป็นแบบนั้น
และเราจะทำระหว่าง x เท่ากับ 0 
กับ x เท่ากับ 8
และคุณจะได้รูปแบบนี้
และคุณหาปริมาตรรูปนี้ได้
ง่ายๆ โดยใช้วิธีแบบจาน
แต่เพื่อแสดงให้คุณดูว่าคุณทำอีกวิธีได้
เราจะใช้วิธีแบบเปลือก
เราจะใช้วิธีแบบเปลือกตอนนี้
หมุนรอบเส้นแนวนอน กล่าวให้ชัดคือ
แกน x
แล้วเราทำได้อย่างไร?
สิ่งที่เราอยากทำ สิ่งที่คุณนึกได้
คือสร้างสี่เหลี่ยมมุมฉากที่เป็นแบบนี้
ขอผมใช้สีส้มแซลมอนนะ
คุณมีสี่เหลี่ยมมุมฉากเป็นแบบนี้
ความหนาหรือความสูงของมัน คือ dy
แล้วความยาวของมันตรงนี้

Korean: 
이 동영상에서는 함수
이 동영상에서는 함수
y=x⅓을
x축 주위로 회전시키겠습니다
그것을 하면
우리는 회전체를 얻게 됩니다
그것은 이렇게 생겼습니다
이것을 x=0부터 x=8까지 할 것 입니다
그럼 이렇게 생기게 됩니다
이것을 디스크 법으로 적분하면
꽤 쉽게 이것의 부피를 찾을 수 있습니다
하지만 여러분들에게 
다른 방법이 있다는 것을 보여주기 위해
원통셸 적분을 하겠습니다
하지만 여기서 우리는
셸 적분을 수평축에 대해 
회전하여 사용해야합니다
x축에 대해서 말입니다
이것을 어떻게 해야할까요
우리가 해야할 것은
뭐 상상을 해 봅시다
이렇게 생긴 직사각형을 만듭니다
이것을 연어색으로 칠해 보겠습니다
그럼 이런 직사각형이 생기게 됩니다
이것의 높이는 dy라고 말할 수 있습니다
그리고 여기 이것의 길이는

Portuguese: 
Neste vídeo faremos a rotação da função
y igual a raiz cúbica de x em torno
do eixo x.
Ao fazermos isso teremos um sólido de 
revolução parecido com este.
Trabalharemos no intervalo de x igual a 0
até x igual a oito,
e teremos uma figura parecida com esta.
Poderíamos usar o método do disco neste
problema, mas como alternativa usaremos
o método das cascas, usando a rotação
em torno do eixo x.
E como faremos isso?
Bem, como você pode supor, começaremos
construindo um retângulo parecido com este
desenhado em cor salmão.
Sua profundidade, ou altura, será dy.
E seu comprimento será

Bulgarian: 
В това видео ще вземем 
функцията
у е равно на корен трети от х
и ще я завъртим
около оста х.
Получаваме ротационно тяло,
което изглежда ето така.
Правим това в интервала между
х = 0 и х = 8.
Получава се нещо такова.
Можем да намерим обема
на това тяло
много лесно с метода на
кръговете.
Но искам да ти покажа, че
има и алтернативен начин,
като ще използваме метода
на черупката.
Сега ще използваме
метода на черупката,
за ротация около хоризонталната
ос, която по-точно е оста х.
Как ще направим това?
Искам да си представим,
че построяваме правоъгълник,
който изглежда ето така.
Ще го направя с този
цвят на сьомга.
Имаме правоъгълник, който
изглежда ето така.
Неговата дълбочина или височина,
можем да кажем, е dy.
Дължината ето тук

Korean: 
8-x값이 됩니다
확실히 해 봅시다
확실히 해 봅시다
이것의 폭은 8-x입니다
x는 여기있는 이 값입니다
그리고 이미 알아챘겠지만
이것은 dy가 됩니다
우리는 y에 대해 적분할 것입니다
y의 구간에서요
그렇기 때문에 우리는 모든 함수를 
y에 대해 나타내야 합니다
이 x값은 y에 대한 함수로 바뀌어야 합니다
y=x⅓일 때
우리는 등식 양 쪽에 세제곱을 해줌으로써
x=y³이라는 식을 얻을 수 있습니다
이 식은 이것과 동일한 식입니다
그러면 이 거리는 y에 대해 표현하면
8-y³이 되어야 합니다
이제 이것을 x축으로 회전시킵니다
이것은 원통의 바깥부분
혹은 우리가 부르기 좋아하는
원통껍질이 됩니다
최선을 다해서
원통 껍질을 그려보겠습니다

English: 
is going to be 8 minus
whatever x value this is.
Let me make this clear.
The width is going to
be 8 minus whatever
x value this is right over here.
And you might already realize
that if this is going to be dy,
we're going to be integrating
with respect to y over
an interval of y.
And so we really want to have
everything in terms of y.
So this x value, whatever
it is, as a function of y.
So if y is equal to
the cube root of x,
we can cube both
sides, and we can
get x is equal to y
to the third power.
These are equivalent
statements right over here.
And so, this distance right over
here is going to be 8 minus y
to the third, if we were to
express it in terms of y.
And then when you rotate
that thing around the x-axis,
it's going to construct
the outside of the cylinder
or shell, as we like to call it.
And I'll try my best
to draw that shell.

Portuguese: 
oito menos qualquer valor inicial de x.
Escrevendo melhor, temos
a largura será 8 menos o valor inicial de x
E, como você pode imaginar, este será dy.
Faremos a integração em relação a y
para um determinado intervalo de y.
Teremos todos os componentes
em função de y.
Assim, este valor de x, qualquer que seja,
será uma função de y.
Se y é igual à raiz cúbica de x,
elevando os dois lados ao cubo
teremos x igual a y ao cubo.
Estas são sentenças equivalentes, dessa
forma a distância aqui representada será
8 menos y ao cubo, se expressarmos em
função de y.
E quando efetuamos a rotação em torno do 
eixo x,
construiremos a parte externa de um
cilindro, ou casca.
Farei o melhor desenho possível,

Thai: 
จะเป็น 8 ลบค่า x ใดก็ตามตรงนั้น
ขอผมบอกให้ชัดนะ
 
ความกว้างจะเท่ากับ 8 ลบค่า x
ใดก็ตามที่อยู่ตรงนี้
แล้วคุณอาจสังเกตได้ว่า ถ้านี่คือ dy
จะอินทิเกรตเทียบกับ y
ตลอดช่วงของ y
และเราอยากให้ทุกอย่างอยู่ในรูปของ y
ค่า x นี้ ไม่ว่ามันจะเป็นอะไร เป็นฟังก์ชันของ y
ถ้า y เท่ากับรากที่สามของ x
เราก็กำลังสามทั้งสองด้านได้ และเรา
จะได้ x เท่ากับ y กำลังสาม
พวกนี้คือประโยคที่เทียบเท่ากัน
แล้ว ระยะนี่ตรงนี้จะเท่ากับ 8 ลบ y
กำลังสาม ถ้าเราเขียนมันในรูปของ y
แล้วเมื่อคุณหมุนรูปนี้รอบแกน x
มันจะสร้างผิวนอกทรงกระบอก
หรือเปลือก แล้วแต่เราจะเรียก
และผมจะพยายามวาดเปลือกนั้นให้ดีที่สุด

Bulgarian: 
е 8 минус стойността на х,
каквато и да е тя.
Нека да поясня.
Широчината ще бъде 8 минус
стойността на х ето тук.
И може би вече разбираш, че
ако това е dy,
то ще интегрираме спрямо у
в интервал за у.
Ще изразим всичко чрез у.
Тази стойност на х, каквато
и да е тя, като функция от у.
Ако у е равно на корен
трети от х,
можем да повдигнем на куб
двете страни и ще получим
х = у^3.
Тези двете са 
еквивалентни изрази.
Това разстояние ето тук
е равно на 8 – у^3,
ако искаме да го изразим
чрез у.
После можем да завъртим
около оста х,
и ще получим външността
на един цилиндър
или черупка, ако предпочиташ 
да го наречеш така.
Ще се постарая да начертая
тази черупка.

Bulgarian: 
Ще изглежда нещо такова.
Получаваме черупка, която
изглежда приблизително така.
Ето това е.
Всъщност тук има обща
граница ето тук.
Ще имаме черупка, която 
изглежда подобно на това.
Надявам се, че това
е от помощ.
Сега ще добавя малко 
дебелина.
Черупката ще изглежда
подобно на това.
Ако можем да намерим
обема на тази черупка,
това ще е равно на лицето
на околната повърхнина
по дебелината.
И после ще сумираме всички
тези черупки,
тази черупка съответства
на дадено у в нашия интервал.
Ако сумираме за всяко
у в нашия интервал
обемите на тези черупки, тогава
ще получим обема 
на това тяло.
Как ще намерим обема
на тази черупка?
Можем да намерим 
обиколката,

Thai: 
มันจะเป็นแบบนี้
 
คุณจะได้เปลือกที่เป็นแบบนี้
ได้แล้ว
ที่จริง มันมีขอบร่วมกัน
ตรงนี้
คุณจะได้เปลือกที่เป็นแบบนี้
หวังว่ามันคงช่วยได้หน่อย
ขอผมใส่ความหนาหน่อย
 
คุณจะมีเปลือกที่เป็นแบบนั้น
ถ้าคุณหาปริมาตรของเปลือกนั้น
ซึ่งก็คือพื้นที่เปลือกนอก
คูณความหนา
แล้วถ้าเรารวมเปลือกทั้งหมดนี้เข้าด้วยกัน
เปลือกนี้ก็คือ y ในช่วงของเรา
ถ้าเราบวกค่า y ทั้งหมดในช่วงนี้
ปริมาตรเปลือกทั้งหมด แล้วเรา
จะได้ปริมาตรของรูปนี้
ย้ำอีกครั้ง เราหาปริมาตรเปลือกได้อย่างไร?
เราหาเส้นรอบวง

Korean: 
그럼 이렇게 생기게 됩니다
그럼 이렇게 생기게 됩니다
그럼 이렇게 생긴 원통껍질이 생깁니다
좋습니다
실제로 이것은 경계를 공유합니다
바로 이곳에서요
여러분들은 이렇게 생긴 원통 껍질을 갖게 됩니다
조금이라도 도움이 되었으면 좋겠습니다
음양을 표현해 보겠습니다
음양을 표현해 보겠습니다
이렇게 생긴 원통 껍질을 갖게됩니다
만약 이 원통 껍질의 
부피를 유추한다면
바깥 쪽 넓이에
그 깊이를 곱한 값이 될 것 입니다
그리고 이러한 원통껍질의 부피를 모두 더한다면
이 껍질은 우리 적분에
특정한 y값에 대한 껍질입니다
우리가 구간 내 모든 y에 대한
원통 껍질의 부피를 더한다면
이 물체의 전체 부피를 구할 수 있습니다
다시 한번 우리는 어떻게 원통 껍질의
부피를 구할 수 있을까요
우리는 둘레를 구할 수 있습니다

Portuguese: 
ficará parecido com este aqui.
Bem, vamos lá...
eles dividem uma borda bem aqui
e ficaremos com uma casca parecida com esta.
Espero que ajude um pouco.
Vamos adicionar um pouco de profundidade,
e a casca ficará parecida com isto.
Podemos determinar o volume da casca,
que será igual à área da superfície
externa vezes a profundidade.
E então somamos todas as cascas, pois
a que desenhamos é apenas uma casca
em particular.
Efetuando a soma de todos os volumes
das cascas presentes no intervalo de y
desejado, então temos o volume da figura.
Retomando, como calculamos o 
volume de uma casca cilídrica?
Primeiro determinamos o comprimento da

English: 
So it will look
something like this.
So you're going to have a
shell that looks like this.
And there you go.
And actually, this shares
a common boundary right
over here.
So you're going to have a shell
that looks something like that.
So that hopefully
helps a little bit.
Let me give it some depth.
You'll have a shell that
looks something like that.
If we could figure out
the volume of that shell,
which is really going to be
the area of the outer surface
area times the depth.
And then if we were to sum
up all of those shells,
this shell is for a
particular y in our interval.
If we were to sum up over all
of the y's in our interval,
all the volumes of
the shells, then we
have the volume of this figure.
So once again, how do we figure
out the volume of a shell?
Well, we can figure
out the circumference

Portuguese: 
circunferência à esquerda ou à direita
de nosso cilindro.
Esse comprimento será calculado com
duas vezes pi vezes o raio.
E o que é o raio?
O raio desses cilindros será o 
nosso valor de y,
e será a distância representada aqui.
O comprimento será então 
2 pi vezes y.
E a área da superfície externa será
calculada multiplicando o comprimento da
circunferência pela largura do cilindro.
Vamos escrever tudo agora.
A área da superfície externa será dada por
2 pi vezes y vezes a diferença
entre oito e y ao cubo.
oito menos y ao cubo equivale 
à extensão.
Multiplicamos a extensão pelo
comprimento da circunferência.

Thai: 
ทางซ้ายหรือทางขวา ในกรณีนี้ --
ทางซ้ายหรือขวาของทรงกระบอก
ถ้าเราหาเส้นรอบวง
เส้นรอบวงเท่ากับ 2 พายคูณรัศมี
แล้วรัศมีคืออะไร?
รัศมีของตัวนี้จะเท่ากับค่า y ของคุณ
นั่นคือระยะนี่ตรงนี้
มันจะเท่ากับค่า y
มันจะเท่ากับ 2 พายคูณ y
แล้วถ้าเราต้องการพื้นที่ของเปลือกนอก
เราก็แค่คูณเส้นรอบวง
ด้วยความกว้างของทรงกระบอก
 
ขอผมเขียนมันตรงนี้นะ
พื้นที่ผิวด้านนอกเท่ากับ 2 พาย
y คูณ 8 ลบ y กำลังสาม
 
8 ลบ y กำลังสองก็แค่ความยาว
คุณคูณมันกับเส้นรอบวง

Bulgarian: 
например на лявата или на дясната,
в този случай... на лявата
или на дясната основа 
на нашия цилиндър.
Ако намерим обиколката,
тази обиколка е равна на
2π по радиуса.
Колко е радиусът?
Радиусът на тези е просто
стойността на у.
Това е това разстояние
ето тук.
Това е просто стойността на у.
Това е равно на 2π по у.
После, за да намерим
околната повърхнина,
само умножаваме обиколката
по широчината на цилиндъра.
Ще го запиша тук.
Лицето на околната повърхнина
е равно на
2π по у по нашето (8 – у^3).
(8 – у^3) е просто тази 
дължина.

Korean: 
여기서는 왼쪽과 오른쪽의
둘레를 구해야겠습니다
원통의 왼쪽이나 오른쪽
우리가 이 둘레를 구한다면
둘레는 2πr입니다
반지름은 무엇입니까
반지름은 y값과 동일합니다
그 거리는 여기 있습니다
y값이 됩니다
그럼 둘레는 2πy가 됩니다
그리고 우리가 바깥 면적을 원한다면
원통의 폭과 둘레를
곱하면 됩니다
곱하면 됩니다
여기에 써 보겠습니다
바깥 면적은
2πy(8-y³)이 됩니다
2πy(8-y³)이 됩니다
8-y³는 이 길이입니다
여러분들이 이것에 둘레를 곱해주면

English: 
of I guess the left or the
right, in this case-- the left
or the right of our cylinder.
If we can figure out
that circumference,
that circumference is equal
to 2 pi times the radius.
And what is the radius?
The radius of these things are
just going to be your y value.
That's this distance
right over here.
That's just going
to be the y value.
So it's going to be
equal to 2 pi times y.
And then if we want the
area of the outer surface,
we just multiply
the circumference
times the width of our cylinder.
So let me write it over here.
So outer surface area is
going to be equal to our 2 pi
y times our 8 minus
y to the third.
8 minus y to the third
is just this length.
You multiply that
times circumference.

English: 
You get the outer surface area.
Now, if you want to find the
volume of this one shell,
it's going to be the
outer surface area.
2 pi y times 8 minus y to
the third times the depth,
times this dy right over here.
I'll do the dy in purple.
So that's the
volume of one shell.
If we want to find the volume of
the entire solid of revolution,
we have to sum all these up
and then take the limit as they
become infinitely
thin, and we have
an infinite number
of these shells.
So we're going to take the
sum from, and remember,
we're dealing in y.
So the volume is
going to be equal to,
so what's our interval
in terms of y?
So y definitely starts off at
0, and when x is equal to 8,
what is y?
Well, 8 to the 1/3
power is just 2.
So y is 2, this value
right over here.
Let me make it a
little bit clearer.
This value right over here is 2.

Bulgarian: 
Умножаваме по обиколката
и получаваме околната повърхнина.
За да намерим обема
на тази черупка,
това е околната повърхнина.
2πу по (8 – у^3), и умножаваме
по дебелината,
по това dy ето тук.
Ще направя dy в цикламено.
Това е обемът на черупката.
Ако искаме да намерим обема
на цялото ротационно тяло,
трябва да сумираме тези и
после да намерим границата, когато
те стават безкрайно тънки и имаме 
безкрайно голям брой черупки.
Значи ще сумираме от...
спомни си, че работим по у.
Значи обемът ще е равен на –
какъв е интервалът
спрямо у?
у определено започва от 0
и после, когато х е равно на 8,
колко е у?
8 на степен 1/3 е просто 2.
Значи тази стойност тук е 2.
Нека да го разясня по-добре.
Стойността ето тук е 2.

Portuguese: 
Obtendo assim a área da superfície.
Para calcularmos o volume dessa casca,
faremos o produto entre a área da casca e
a profundidade, ou seja
2 pi vezes y vezes a diferença 
entre oito e y ao cubo vezes dy
Esse será o volume de uma casca.
Para determinar o volume do sólido 
de revolução completo
devemos somar todas as cascas e 
tomar o limite até que elas
se tornem infinitesimalmente finas, e
adotando uma quantidade infinita de cascas
Não se esqueça que estamos somando
em relação a y.
Então vamos determinar o intervalo
em relação a y.
O valor inicial certamente é zero, e
para x igual a 8, qual será o valor de y?
Bem, 8 elevado a 1/3 equivale a dois.
Então, y vale dois.
Deixando um pouco mais evidente,
este valor aqui é 2.

Korean: 
원통 표면의 넓이를 구할 수 있습니다
만약 이 원통 껍질의 부피를 구하려면
표면 넓이인
2πy(8-y³)에 두께를 곱합니다
두께는 여기서 dy입니다
dy를 보라색으로 쓰겠습니다
dy를 보라색으로 쓰겠습니다
이것이 한 원통껍질의 부피입니다
만약 우리가 전체 회전체의
부피를 구하고 싶다면
우리는 이 모든 것을 더한 후
그들의 두께가 무한히 얇아지도록
원통 껍질의 수를 무한히 늘려야 합니다
그러면 이들을 더해보겠습니다
우리는 y에 대해 계산하고 
있다는 것을 유의합시다
그럼 부피는...
우리가 구하는 y의 구간은 
어디입니까
y는 확실히 0에서 시작합니다
x가 8이 될 때
y의 값이 무엇입니까
8⅓는 2입니다
그럼 여기 y값은 2입니다
조금 더 정확히 그려보겠습니다
여기의 값은 2입니다

Thai: 
คุณจะได้พื้นที่ผิวด้านนอก
ทีนี้ ถ้าเราอยากหาปริมาตรของเปลือกนี้
มันจะเป็นพื้นที่ผิวด้านนอก
2 พาย y คูณ 8 ลบ y กำลังสาม คูณความหนา
คูณ dy นี่ตรงนี้
ผมจะเขียน dy สีม่วง
 
นั่นคือปริมาตรของเปลือก
ถ้าเราอยากหาปริมาตรของทรงตันทั้งรูป
เราต้องบวกค่าเหล่านี้ทั้งหมด แล้วหาลิมิตเมื่อ
เปลือกบางเฉียบ และเรา
มีจำนวนเปลือกนับไม่ถ้วน
เราจะหาผลบวกจาก นึกดู
เรากำลังคิด y
ปริมาตรจะเท่ากับ
ช่วงในรูปของ y เป็นเท่าใด?
y เริ่มต้องจาก 0 และเมื่อ x เท่ากับ 8
y เป็นเท่าใด?
8 กำลัง 1/3 ก็แค่ 2
y เป็น 2 ค่านี่ตรงนี้
ขอผมบอกให้ชัดขึ้นนะ
ค่านี่ตรงนี้คือ 2

English: 
So y goes from 0 to 2, and
we've set up our integral.
And this one looks
pretty straightforward.
So I think we can crank
through it in this video.
So this is going to be equal
to, we can take out the 2 pi,
2 pi times the definite
integral from 0 to 2.
Let's multiply the y of
8y minus y to the fourth.
All of that dy.
This is going to be equal to
2 pi times the anti-derivative
of this business.
Anti-derivative of
8y is 4y squared.
Anti-derivative of 1
negative y to the fourth
is negative y to
the fifth over 5.
And we're going to
evaluate it at 0 and 2.
So this is going to be
equal to 2 pi times,
we evaluate this business at 2,
2 squared is 4 times 4 is 16.

Korean: 
그럼 적분구간은 0부터 2까지입니다
우리는 적분식을 만들었습니다
그리고 이것은 꽤 간단해보입니다
그래서 우리가 이 동영상에서
이 식을 끝낼 수 있을 것 같습니다
적분식에서 2π를 꺼낼 수 있습니다
2π*0에서 2까지 정적분
y를 곱해 8y-y^4입니다
이것 전체에 dy를 곱합니다
이것 전체에 dy를 곱합니다
이것은 2π에 모든 적분한 값을
곱한 것과 같습니다
8y의 역도함수는 4y²입니다
8y의 역도함수는 4y²입니다
(-y^4)의 부정적분은
-(1/5)y^5입니다
이것을 0부터 2까지 계산하면
2π에
2를 대입한 값을 곱합니다 4*2^2=16

Thai: 
y ไปจาก 0 ถึง 2 และเราตั้งอินทิกรัลได้
อันนี้ค่อนข้างตรงไปตรงมา
ผมว่าเราทำในวิดีโอนี้ได้เลย
อันนี้จะเท่ากับ เรานำ 2 พายออกมาได้
2 พายคูณอินทิกรัลจำกัดเขตจาก 0 ถึง 2
ลองคูณ y ได้ 8y ลบ y กำลัง 4
ทั้งหมดนั้น dy
 
อันนี้จะเท่ากับ 2 พายคูณปฏิยานุพันธ์
ของตัวนี้
ปฏิยานุพันธ์ของ 8y คือ 4y กำลังสอง
 
ปฏิยานุพันธ์ของ 1 ลบ y กำลังสี่
เป็นลบ y กำลังห้าส่วน 5
และเราจะหาค่ามันที่ 0 กับ 2
อันนี้จะเท่ากับ 2 พายคูณ
เราหาค่าตัวนี้ที่ 2, 2 กำลังสองได้ 4 
คูณ 4 เป็น 16

Portuguese: 
Então y varia de 0 a 2, e podemos
escrever nossa integral.
A integral que aparece é bem direta,
então vamos ao trabalho.
Colocando o 2 pi em evidência fora da integral,
temos 2 pi vezes a integral definida entre 0 e 2.
Multiplicando por y temos 8y menos
y à quarta potência. Tudo vezes dy.
e será igual a 2 pi vezes a antiderivada
de tudo isso.
A antiderivada de 8y é 4y ao quadrado.
A antiderivada de menos 1 
vezes y á quarta potência
é menos y elevado à quinta
potência sore 5.
E calculando no intervalo entre 0 e 2.
Para y igual a 2, teremos 2 pi vezes
2 ao quadrado (quatro) vezes 4 (dezesseis)

Bulgarian: 
у е от 0 до 2, и нашият
интеграл е готов.
Това изглежда много лесно.
Мисля, че ще успеем
да го довършим в този клип.
Това ще е равно на...
изнасяме пред интеграла 2π,
2π по определен интеграл
от 0 до 2.
Сега да умножим с у, 
получаваме 8у – у^4.
Цялото това по dy.
Това е равно на 2π
по примитивната функция
на този израз.
Примитивната функция на
8у е 4у^2.
Примитивната функция на –у^4
е –у^5 върху 5.
И ще изчислим това за 0 и за 2.
Това ще е равно на 2π по...
изчисляваме това за 2,
2^2 е 4, по 4 е 16.

Korean: 
2^5=32
16-32/5
16-32/5
0을 대입하면 그냥 0이 나옵니다
이게 우리에게 남은 것입니다
이제 이것들을
조금 단순히 하겠습니다
자
16과 1/5 여기에 쓰겠습니다
16과 1/5 여기에 쓰겠습니다
16은 80/5와 동일합니다
여기서 32/5를 뺍니다
여기서 32/5를 뺍니다
이것은 48/5같습니다
이 식의 결과는 48/5입니다
제가 한 것이 맞나요
80-30-2=48
그럼 48/5*2π
이제 드럼연주가 필요하겠습니다
48*2=96
이것을 새로운 색으로 칠하겠습니다
우리가 다 했다는 것을 강조하기 위해서요
답은 96π/5입니다

Thai: 
16 ลบ 2 กำลังห้าเป็น 32
ลบ 32/5
 
แล้ว คุณหาค่ามันที่ 0 คุณจะได้ 0
นั่นคือสิ่งที่เราเหลือ
แล้วตอนนี้ เราต้องจัดรูป
ตัวนี้หน่อย
ลองดูกัน
16 ส่วน 5 ส่วนนี่ตรงนี้
 
16 เท่ากับ จะเรียกว่า 80/5 ก็ได้
แล้วจากนั้น เราจะลบ 32/5
 
แล้ว อันนี้เท่ากับ 48/5
ทั้งหมดนี้จึงเท่ากับ 48/5
ผมทำถูกไหม?
80 ลบ 30 ได้ 50, แล้วลบอีก 2 ได้ 48
นี่คือ 48/5 คูณ 2 พาย
ตอนนี้เราพร้อมตีกลองต้อนรับแล้ว
48 คูณ 2 ได้ 96
ขอผมใช้สีใหม่นะ
เพื่อเน้นว่าเราเสร็จแล้ว
48 คูณ 2 ได้ 96 คูณพายส่วน 5

Bulgarian: 
16 –... 2^5, което е 32.
Значи минус 32/5.
После изчисляваме това
за 0, което е просто 0.
Значи получаваме това.
И сега само трябва малко
да го опростим.
16 върху 5, тази част ето тук.
Извинявам се, 
16 е равно на 80/5.
От това вадим 32/5.
Получаваме 48/5.
Цялото това нещо е
равно на 48/5.
Вярно ли го сметнах?
80 минус 30 е 50, после
минус 2 става 48.
Това е 48/5 по 2π.
И сега заслужаваме 
аплодисменти.
48 по 2 е 96, или
получаваме 96 по пи върху 5.
Ще използвам нов цвят,
за да подчертаем, че
сме на финала.
48 по 2 е 96 по π/2.

English: 
16 minus 2 to the fifth is 32.
So minus 32/5.
And then, you evaluate this
stuff at 0, you just get the 0.
So that's what we're left with.
And now, we just
have to simplify
this thing a little bit.
So let's see.
16 over 5, this part
right over here.
16 is the same thing,
I should say, as 80/5.
And from that, we
are subtracting 32/5.
And so, that is equal to 48/5.
So all of this business
is equal to 48/5.
Did I do that right?
80 minus 30 is 50, and then
minus another 2 gets us to 48.
So this is 48/5 times 2 pi.
And now we deserve
our drum roll.
48 times 2 is 96.
Let me do this in
a new color, just
to emphasize that
we're at the end.
48 times 2 is 96
times pi over 5.

Portuguese: 
menos 2 elevado à quinta potência
sobre 5, ou seja, menos 32/5.
E para y igual a zero, toda a 
expressão vale zero.
Ficamos então com esta expressão.
Agora devemos simplificar um pouco
as coisas, observe.
16 equivale a 80 sobre 5
E dele, subtraímos 32 sobre 5
Restando 48 sobre 5
Então toda essa parte vale 48 sobre 5,
de acordo?
80 menos 30 vale 50, menos dois, 48.
Então temos 48 sobre 5 vezes 2 pi.
E agora, finalizando (e que rufem os tambores)
48 vezes dois é igual a 96.
Mudarei de cor para enfatizar que
estamos no final.
48 vezes dois é 96 vezes pi sobre 5.

Bulgarian: 
Повтарям отново, това е 
нещо, което
можеш да решиш по метода на
дисковете спрямо х.
Но тук показах, че можеш
да го решиш и с помощта
на метода на черупката по
отношение на у.
Метод на черупката или 
на кухия цилиндър,
както си избереш, 
по отношение на у.

English: 
So once again, this
is something that you
could have solved using the
disk method in terms of x.
And we're just showing that
you could also solve it
in the shell method
in terms of y.
The shell or the hollowed
out cylinder method,
whatever you want to
call it, in terms of y.

Thai: 
ย้ำอีกครั้ง นี่คือสิ่งที่คุณ
ทำได้โดยใช้วิธีแบบจานในรูปของ x
และเราเพิ่งแสดงว่า คุณแก้ได้
ด้วยวิธีแบบเปลือกในรูปของ y เช่นกัน
วิธีแบบเปลือกหรือทรงกระบอกกลวง
แล้วแต่คุณจะเรียกว่าอะไร ในรูปของ y

Korean: 
다시 말하지만 이 문제는
x에 대한 디스크 적분법으로도 
해결할 수 있습니다
그리고 우리는 y에 대한 
원통셸 적분법으로도
문제를 풀 수 있다는 것을 방금 보였습니다
셸 적분법 또는 속이 빈 원통 적분법
이것을 뭐라고 부르든지 간에요
커넥트 번역 봉사단 | 김현수

Portuguese: 
Então, mais uma vez, aqui temos algo que
poderia ter sido
resolvido usando o método do 
disco em função de x.
Estamos apenas mostrando que o problema
também pode ser solucionado
usando o método das conchas
em função de y.
Legendado por [Tatiana F. D'Addio]
