
Thai: 
 
ตรงนี้ผมได้วาดส่วนหนึ่งของกราฟ y
เท่ากับ x กำลังสอง
และสิ่งที่เราจะทำคือใช้พลัง
ของอินทิกรัลจำกัดเขตหาปริมาตร 
แทนที่จะเป็นแค่พื้นที่
ลองทบทวนสิ่งที่เราจะทำเวลา
เราหาอินทิกรัลจำกัดเขตทั่วไปกัน
ถ้าเรามีอินทิกรัลจำกัดเขตระหว่าง อย่างเช่น
0 ถึง 2 ของ x กำลังสอง dx มันจะแทนอะไร?
ลองดูที่จุดปลายของเรากัน
นี่คือ x เท่ากับ 0
สมมุติว่าจุดนี่ตรงนี้คือ x เท่ากับ 2
สิ่งที่เราจะทำคือสำหรับ x แต่ละตัว เรา
จะหา dx เล็กๆ รอบมัน -- เส้นนี่
ตรงนี้คือ dx เล็กๆ
และเราจะคูณ dx นั่นด้วยฟังก์ชันของเรา
คูณ x กำลังสอง
สิ่งที่เราจะทำคือเราจะคูณความกว้างนี้ด้วย
ความสูงนี่ตรงนี้
ความสูงตรงนี้คือ x กำลังสอง
และเราจะได้พื้นที่ของสี่เหลี่ยมมุมฉากเล็กๆ นี้

Czech: 
Zde jsem nakreslil část grafu funkce
y se rovná 'x na druhou'.
Teď použijeme znalosti určitých integrálů
k výpočtu objemu, nejen plochy.
Zopakujme si teď, jak postupovat
při výpočtu obyčejného určitého integrálu.
Vezmeme-li například určitý integrál
od 0 do 2 z 'x na druhou', co to znamená?
Podívejme se na meze.
V dolní mezi je x rovno 0
a řekněme, že v tomto místě 
bude x rovno 2.
Integrací děláme to, že ke každému x
vezmeme malou úsečku ‚dx‘…
Řekněme, že toto je malé ‚dx‘…
Tuto úsečku násobíme funkční hodnotou,
tedy krát 'x na druhou'.
Tuto úsečku o šířce ‚dx‘
násobíme právě touto úsečkou,
která má výšku 'x na druhou'.
Tím získáme obsah tohoto úzkého obdélníku.

Bulgarian: 
 
Тук съм начертал част
от графиката на функцията
у = x^2.
Сега ще използваме нашите
знания за определените интеграли,
за да намираме обеми,
а не само площ.
Да преговорим 
какво правим, когато
имаме един определен интеграл.
Нека да имаме определен
интеграл между, да кажем
между 0 и 2 от x^2dх,
какво представлява това?
Тези изглеждат като крайни точки.
Това е х = 0.
Това тук е х = 2.
За всяка стойност на х
ние намираме
едно малко dх, около него,
така че тук имаме малко dх.
И умножаваме това dх
по нашата функция,
по х^2.
Значи тук умножаваме тази
широчина
по тази височина ето тук.
Височината тук е х^2.
И получаваме площта
на този малък правоъгълник.

Spanish: 
Aquí dibuje una parte de la 
gráfica de la función y
igual a x al cuadrado (y=x^2)
Vamos a usar nuestros conocimientos de
integrales definidas para encontrar 
volúmenes en lugar de áreas
Revisemos lo que 
usualmente hacemos
cuando evaluamos integrales definidas.
Tomemos la intergal definida, 
digamos entre
0 y 2 de x al cuadrado dx,
¿qué es lo que representa?
Veamos el intervalo [0,2]
Aquí tenemos cuando x es igual a 0
Y digamos que aquí es
cuando x es igual a 2
Lo que hacemos es, para cada x
encontramos un pequeño dx alrededor de x, 
--esto de aquí es
el pequeño dx--
Y multiplicamos ese dx por 
nuestra función,
es decir por x^2.
Lo que hacemos es multiplicar 
este ancho por
por esta altura de acá.
La altura aquí es 
precisamente x al cuadrado
Lo que obtenemos es el área de este
pequeño rectángulo.

Korean: 
여기에 있는 그래프는
여기에 있는 그래프는
y = x²의 그래프입니다
이번 동영상에서 해 볼 것은
정적분을 이용해
부피를 계산하는 것입니다
그 전에 일반적인 정적분이
무엇을 계산하는 것인지를
다시 복습해보도록 하겠습니다
0에서 2까지 x²의 정적분이
의미하는 바는 무엇입니까?
이 적분의 범위는
x = 0에서부터
x = 2까지입니다
이 범위 내의 각각의 x에 대해
아주 작은 변화량인
dx를 생각해봅시다
dx를 그 위치에서의 함숫값인
x²과 곱하면
이 직사각형에서의
밑변과 높이를 곱한 것이므로
방금 계산한 값은
이 작은 직사각형의 넓이가 됩니다

English: 
Over here I've drawn
part of the graph of y
is equal to x squared.
And what we're going
to do is use our powers
of definite integrals to find
volumes instead of just areas.
So let's review what
we're doing when
we take just a regular
definite integral.
So if we take the definite
integral between, say,
0 and 2 of x squared dx,
what does that represent?
Well, let's look
at our endpoints.
So this is x is equal to 0.
Let's say that this right
over here is x is equal to 2.
What we're doing is
for each x, we're
finding a little dx around
it-- so this right over here
is a little dx.
And we're multiplying that
dx times our function,
times x squared.
So what we're doing is we're
multiplying this width times
this height right over here.
The height right over
here is x squared.
And we're getting the area
of this little rectangle.

Portuguese: 
Vemos aqui parte do 
gráfico de y
é igual a x quadrado,
e o que estamos querendo
fazer é usar o poder
das integrais definidas para
achar volumes ao invés só de áreas.
Vamos lembrar o que 
fazemos quando
calculamos uma integral
definida regular.
Se, tomarmos a integral definida
entre, digamos,
zero a dois de x ao quadrado dx,
o que isto representa?
Vamos dar uma olhada
nos pontos limites.
Isto é x igual a zero,
e isto bem aqui, digamos, é
x igual a dois.
O que fazemos aqui,
para cada x,
é encontrar um pequeno dx
em torno dele -- então este bem aqui
é um destes pequenos dx.
E multiplicamos este dx
pela nossa função,
vezes x quadrado.
Então, o que estamos fazendo
é multiplicar esta largura
por esta altura bem aqui.
A altura é igual
a x quadrado,
e calculamos a área
deste pequeno retângulo.

Chinese: 
mo
mo
hello
a
a

English: 
And the integral
sign is literally
the sum of all of these
rectangles for all of the x's
between x is equal to
0 and x is equal to 2.
But the limit of that as these
dx's get smaller and smaller
and smaller, get
infinitely small,
but not being equal to 0.
And we have an infinite
number of them.
That's the whole power
of the definite integral.
And so you can imagine, as these
dx's get smaller and smaller
and smaller, these rectangles
get narrower and narrower
and narrower, and we
have more of them,
we are getting a better
and better approximation
of the area under the
curve until, at the limit,
we are getting the
area under the curve.
Now we're going to apply
that same idea, not
to find the area
under this curve,
but to find the
volume if we were
to rotate this curve
around the x-axis.
So this is going to stretch our
powers of visualization here.
So let's think
about what happens
when we rotate this
thing around the x-axis.
So if we were to rotate
it, and I'll look at it

Korean: 
적분 기호가 의미하는 바는
x가 0 이상 2 이하인 범위에서
작은 직사각형의 넓이를
모두 더하는 것이고
dx의 길이는 계속 짧게 만들어서
0은 아니지만 0에 가까워지도록
만들어주면
무한개의 직사각형의
넓이의 합을 얻겠죠
그것이 이 정적분의 의미입니다
dx의 길이가 점점 짧아지므로
각각의 직사각형은 점점 얇아집니다
동시에 직사각형의 개수도
점점 많아질 것이므로
이 값은 곡선과 x축 사이 넓이의
더 좋은 근삿값이 되겠죠
결국 곡선과 x축 사이의
넓이를 구할 수 있는 것입니다
이번에는 같은 방법을 이용해서
곡선 아래의 넓이를
구하는 것이 아니라
이 곡선을 x축에 대해
회전시켜서 나온 입체도형의
부피를 구해보도록 하겠습니다
우선 이 곡선을 x축에 대해
회전시켜 얻은 입체도형이
어떻게 생겼는지

Portuguese: 
Esta integral é, 
literalmente,
a soma de todos estes retângulos
para todos os x
entre zero e
x igual a dois.
No limite, estes dx
ficam menores, menores
e menores, infinitamente
pequenos,
mas não iguais à zero
e temos um número
infinito deles.
Este é o poder
da integral definida,
você pode imaginar, conforme
estes dx ficam menores e menores
e menores, estes retângulos mais
e mais e mais
estreitos, e quanto mais
retângulos tivermos
melhor e melhor será
a aproximação
para a área sob esta
curva, até termos,
no limite, a área
sob a curva.
Agora vamos aplicar
a mesma ideia,
não para encontrar a área
sob esta curva,
mas para encontrar
o volume
se rotacionarmos esta curva
em torno do eixo x.
Isto deve melhorar nossas
habilidades de visualização.
Vamos pensar sobre o que
acontece
quando rotacionamos isto aqui
em torno do eixo x.
Se vamos rotacionar isto
-- deixe-me ver isto e,

Spanish: 
El simbolo de la integral 
es literalmente
la suma de todos los 
rectángulos de las demás x´s
dentro del intervalo [0,2]
Pero, tomando el limite cuando 
esos dx's se vuelven más y más pequeños
y más pequeños, infinitamente pequeños
pero nunca valen 0.
Tenemos una cantidad infinita de ellos
Ese es todo el poder 
de la integral definida
Como se podrán imaginar, 
mientras los dx's se hacen más
y más pequeños, los rectángulos
se hacen más
y más angostos, entre más
rectángulos, 
obtenemos una mejor aproximación
del área bajo la curva, 
hasta que, en el límite
obtenemos el área bajo la curva.
Ahora vamos a aplicar la misma idea, no
para encontrar el área bajo esta curva,
sino para encontrar el volumen del
sólido obtenido al rotar 
la curva alrededor del eje x

Czech: 
Symbol integrálu pak značí
součet všech obdélníčků.
Každý je nad jedním z nekonečně mnoha ‚x‘,
která se nacházejí mezi 0 a 2.
Uvědomme si, že délku ‚dx‘ bereme
menší a menší, až nekonečně malou,
ale nikdy ne nulovou.
Těchto malých úseček
máme nekonečně mnoho.
V tom tkví celá podstata
určitého integrálu.
Představa je taková,
že jak se ‚dx‘ stávají menší a menší,
jsou i obdélníčky nad nimi užší a užší,
přičemž roste i jejich počet.
Čím více jich je, tím blíže je jejich
součet roven ploše pod křivkou,
až v limitě máme přesně
obsah pod křivkou.
Stejnou myšlenku teď uplatníme,
abychom vypočítali
nikoliv obsah,
ale objem tělesa vzniklého
rotací této křivky podél osy x.
To nám procvičí představivost.
Pokuste si představit, co se stane,
bude-li se naše křivka otáčet kolem osy x.
Vytvoří tak plášť tělesa,
na které se díváme mírně zprava.

Bulgarian: 
Знакът за интеграл
просто означава, че
събираме всички тези малки
правоъгълници за всички хиксове
между х = 0 и х = 2.
Но това dх клони към нула,
то е безкрайно малко,
но то не е равно на 0.
И имаме безкрайно много такива.
Това е целият смисъл
на определения интеграл.
Можеш да си представиш, че когато
dх стават все по-малки и по-малки,
тези правоъгълници стават все
по-тесни и по-тесни,
и имаме все повече от тях,
тогава получаваме все 
по-точни и по-точни резултати
за площта под кривата,
между тези граници,
получаваме площта
под кривата.
Сега ще приложим същата
идея,
но не да намираме
площта под крива,
а ще намерим обем, ако
завъртим тази крива около оста х.
Това е изпитание за уменията
да го визуализираме.
Да си представим какво
се случва, когато
завъртим това около
оста х.

Thai: 
และเครื่องหมายอินทิกรัลก็คือ
ผลบวกของสี่เหลี่ยมมุมฉากเหล่านี้ทั้งหมด
สำหรับ x ทุกค่า
ระหว่าง x เท่ากับ 0 กับ x เท่ากับ 2
แต่ลิมิตของมัน เมื่อ dx เหล่านี้เล็กลง เล็กลง
และเล็กลง จนเล็กจิ๋ว
ไม่แต่เท่ากับ 0
เราจะได้จำนวนสี่เหลี่ยมนับไม่ถ้วน
นั่นคือพลังของอินทิกรัลจำกัดเขต
แล้วคุณคงนึกได้ เมื่อ dx เหล่านี้เล็กลง เล็กลง
และเล็กลง สี่เหลี่ยมมุมฉาก
เหล่านี้จะแคบลง แคบลง
และแคบลง เราก็ได้รูปมากขึ้น
เราจะประมาณพื้นที่ใต้เส้นโค้ง
ได้ดีขึ้น ดีขึ้น กระทั่ง ที่ลิมิต
เราจะได้พื้นที่ใต้เส้นโค้งพอดี
ทีนี้ เราจะใช้แนวคิดเดียวกัน ไม่ใช่
เพื่อหาพื้นที่ใต้เส้นโค้งนี้
แต่เพื่อหาปริมาตรถ้าเรา
หมุนเส้นโค้งนี้รอบแกน x
อันนี้จะขยายการมองภาพตรงนี้
ลองคิดกันว่าเกิดอะไรขึ้น
เมื่อเราหมุนรูปนี้รอบแกน x
ถ้าเราหมุนมัน และผมจะดู

Czech: 
Těleso má kruhovou podstavu,
která pak vypadá asi takto.
Lépe to nakreslit neumím.
Máme tedy podstavu
a naše funkce pak tvoří plášť,
funkci uvažujeme jen mezi 0 a 2.
Těleso vypadá jako trychtýř
nebo jako vrchol libereckého Ještědu
nebo jako podivný klobouk.
Ještě tomu dodám trochu stínování,
aby to vypadalo víc trojrozměrně.
Tak nějak tedy vypadá těleso
vzniklé rotací naší křivky.
Nás zajímá celý jeho objem.
Nakreslím ho ještě z jiného úhlu.
Postavené na podstavu vypadá těleso takto.
Takto už skutečně trochu
připomíná podivný klobouk.
Směrem nahoru se špičatí a zakřivuje.
Vypadá nějak takto,
ovšem z tohoto úhlu nevidíme podstavu.

Bulgarian: 
Ако го завъртим и ако 
го гледаме малко отдясно,
тогава основата ще изглежда
горе-долу така.
Ще дам всичко от себе си
да го начертая.
Значи основата изглежда
горе-долу така.
И после останалата
част от функцията,
ако я разгледаме между 0 и 2,
прилича на сплескана
фигурка от игра,
сигурно си играл на играта 
"Не се сърди човече",
това изглежда малко като 
странна шапка.
Изглежда горе-долу така
и сега малко ще оцветя,
значи изглежда като това.
Искам да съм сигурен,
че си представяш как
това нещо се върти тук.
Интересува ни обема
на цялото това нещо.
Ще го нарисувам от няколко
други гледни точки.
Ако го гледаме отгоре,
ще изглежда ето така.
Ще стане по-ясно, че
прилича на шапка.
Сочи нагоре ето така,
и отива надолу ето така.
Ще прилича на нещо такова.

Korean: 
알아보도록 하겠습니다
그 도형의 밑면은
지금 그리고 있는 원 모양으로
나타나게 됩니다
밑면이 아닌
0과 2 사이의 나머지 부분에서는
지금 그리고 있는
모양과 같이 도형이
그려지게 됩니다
명암 표시도 조금 할까요?
이런 입체도형이 완성됩니다
이 동영상에서 구해볼 것은
이 입체도형의
전체 부피입니다
이 도형을 다른 각도에서
그려보도록 하겠습니다
이렇게 그려보니
모자와 비슷한 모양이네요
윗부분은 뾰족하고
아래로 갈수록 단면적이 커집니다

Thai: 
แล้วบอกว่า เรากำลังดูจากทางขวา
เราจะได้ฐานที่ดูเป็นแบบนี้
ผมพยายามวาดให้ดีที่สุดแล้ว
คุณมีฐานที่เป็นแบบนั้น
แล้วฟังก์ชันที่เหลือ
ถ้าเราคิดระหว่าง 0 กับ 2
มันดูเหมือนชิ้นส่วน
จากเกม -- ผมไม่รู้ว่าคุณเคยเล่นเกม 
Sorry หรือเปล่า --
หรือมันดูเหมือนหมวกประหลาดนิดๆ
มันเป็นแบบนี้ ขอผมแรเงาหน่อย
มันจะเป็นแบบนั้น
อย่างนั้น เราจะได้มั่นใจ
ว่าเรามองภาพรูปนี้กำลังหมุนได้
เราสนใจปริมาตรทั้งหมดของรูปนี้
ขอผมวาดมันจากอีกมุมนะ
ถ้าผมวาดจากด้านบน มันจะเป็นแบบนี้
มันจะชัดเจนขึ้นเล็กน้อย
ว่ามันหน้าตาเหมือนหมวก
มันจะชี้ขึ้นแบบนี้ และมันลงมาแบบนั้น
มันจะเป็นแบบนั้น

Portuguese: 
digamos que estejamos
olhando a partir da direita--
teremos uma base
parecida com isto aqui.
--Estou fazendo o meu melhor
para tentar desenhar isto--
Você tem uma base 
parecida com esta,
E o resto será a função,
no intervalo entre
zero e dois,
parecido com uma
peça de ...
-- Não sei se você já jogou
este jogo antes, desculpe--
ou parecido com um
chapéu meio esquisito.
Bem, irá parecer com isto, e
deixe-me sombreá-lo um pouco
para que fique melhor
e tenhamos a certeza
que estamos olhando isto aqui
rotacionado em torno de x.
Estamos preocupados com o
volume total desta forma.
Deixe-me desenhá-la a partir
de diferentes ângulos.
Se desenharmos a partir do topo,
ela irá parecer com isto.
Ficará um pouco
mais óbvio que
parece um chapéu
apontando para cima e
vindo para baixo assim.
Algo deste jeito.

English: 
and say that we're looking it
a little bit from the right.
So we get kind of a base that
looks something like this.
So this is my best
attempt to draw it.
So you have a base that
looks something like that.
And then the rest
of the function,
if we just think
about between 0 and 2,
it looks like one
of those pieces
from-- I don't know if you
ever played the game Sorry--
or it looks like a little
bit of a weird hat.
So it looks like this, and let
me shade it in a little bit
so it looks something like that.
And just so that
we're making sure
we can visualize this
thing that's being rotated.
We care about the entire
volume of the thing.
Let me draw it from a
few different angles.
So if I drew from the top, it
would look something like this.
It'll become a
little more obvious
that it looks
something like a hat.
It would point up like this,
and it goes down like that.
It would look
something like that.

Korean: 
이 방향에서 바라보니
밑면이 보이지 않네요
이 상황에서 각각의 축은 어디에 있을까요?
y축은 이렇게 그려집니다
y축은 이렇게 그려집니다
그리고 x축은 이 도형 안으로
들어가서 반대쪽을 뚫고 나옵니다
만약 이 도형이 투명했다면
이 도형의 뒷부분을 볼 수 있었겠죠
그 뒷부분은 이 점선처럼
생겼을 겁니다
만약 x축도 볼 수 있다면
x축은 도형의 밑면과
이 점에서 교차하여
도형을 뚫고 나오는 모양일 것입니다
이것은 이 도형을 그리는
방법의 하나입니다
다른 여러 방면에서 본
모양으로도 그릴 수 있지요
이 도형의 부피는
어떻게 구할 수 있을까요?
이 작은 직사각형들의 넓이가 아닌
이 직사각형들 각각을 x축에 대해
회전시킨 도형에 대해
살펴봅시다
작은 직사각형 중
여기 dx를 폭으로 하는 직사각형을
x축에 대해 회전시킵니다

Bulgarian: 
От този ъгъл 
не виждаме дъното.
И ако се ориентираш,
в този случай осите
изглеждат ето така.
Това е оста у.
Оста х влиза вътре в това нещо
и после излиза от другата страна.
Ако това е прозрачно,
ще можеш да видиш 
задната му страна.
Ще изглежда ето така.
Оста х, ако можеш 
да виждаш през него,
ще излиза от основата
ето тук,
ще минава през основата
точно ето тук.
И ще излиза от другата страна.
Това са различни ориентации
за едно и също нещо.
Можеш да си го представиш
от различни гледни точки.
Сега да видим как
да сметнем обема.
Вместо да разглеждаме
обема на всички тези
правоъгълници, какво ще стане
ако завъртим всеки
от тези правоъгълници
около оста х?
Да го направим.
Да вземем всеки един от тях.
Да кажем, че тук имам dх,
и го въртим около оста х.

English: 
So in this angle, we're not
seeing the bottom of it.
And if you were to
just orient yourself,
the axes, in this
case, look like this.
So this is the y-axis.
And the x-axis goes right
inside of this thing
and then pops out
the other side.
And if this thing
was transparent,
then you could
see the back side.
It would look
something like that.
The x-axis, if you
could see through it,
would pop the base
right over there,
would go right through
the base right over there.
And it'd come out
on the other side.
So this is one orientation
for the same thing.
You could visualize it
from different angles.
So let's think about how we
can take the volume of it.
Well, instead of thinking
about the area of each
of these rectangles, what
happens if we rotate each
of these rectangles
around the x-axis?
So let's do it.
So let's take each of these.
Let's say you have this
dx right over here,
and you rotate it
around the x-axis.

Portuguese: 
Neste ângulo, não estamos vendo
seu fundo e,
só para nos orientarmos,
os eixos,
ficarão assim.
Este é o eixo y,
e o eixo x vem aqui
por dentro e
aparece do outro
lado.
Se isto fosse transparente
poderíamos ver o outro
lado,
mais ou menos
assim.
O eixo x, se enxergássemos
através disto,
iria furar a base
aqui,
e continuaria
desta forma
até chegar no
outro lado.
Isto é outra vista
da mesma coisa.
Você poderia visualizá-la
de diferentes ângulos.
Vamos pensar sobre como
podemos calcular o seu volume.
Ao invés de pensarmos em termos
da área de cada um
destes retângulos, o que
acontece se rotacionarmos
cada um destes retângulos
em torno do eixo x?
Vamos fazer isto.
Pegamos cada um deles,
digamos que você tenha este
dx bem aqui
e que rotacione em torno
do eixo x.

Thai: 
ในมุมนี้ เราจะไม่เห็นข้างล่าง
ถ้าคุณปรับตัวเอง
แกน ในกรณีนี้ จะเป็นแบบนี้
นี่คือแกน y
 
และแกน x ตรงเข้าไปในรูปทรงนี้
แล้วโผล่ออกมาอีกด้าน
และถ้ารูปนี้โปร่งใส
คุณจะเห็นด้านหลังด้วย
มันจะเป็นแบบนั้น
แกน x ถ้าคุณมองทะลุมันได้
จะแทงฐานออกมาตรงนั้น
ผ่านฐานตรงนั้นพอดี
มันจะออกมาอีกด้านหนึ่ง
นี่ก็คือมุมมองของรูปเดิมอีกมุมหนึ่ง
คุณมองภาพมันจากมุมต่างๆ ได้
ลองคิดดูว่าเราจะหาปริมาตรมันได้อย่างไร
แทนที่จะคิดถึงพื้นที่ของสี่เหลี่ยม
มุมฉากแต่ละรูป เกิดอะไรขึ้น
ถ้าเราหมุนสี่เหลี่ยม
มุมฉากแต่ละรูปรอบแกน x?
ลองทำดู
ลองนำแต่ละรูปมา
สมมุติว่าคุณมี dx นี่ตรงนี้
และคุณหมุนมันรอบแกน x

Czech: 
Jen pro vaši orientaci,
takto vypadají při tomto pohledu osy:
Toto je osa y.
Osa x prochází přímo skrz těleso
a na druhé straně z něho leze ven.
Kdyby bylo těleso průhledné,
byla by vidět zadní strana
a také hrana podstavy.
Osa x by, v případě průhledného tělesa,
pronikala podstavou v tomto místě,
procházela tělesem
a těleso opouštěla ve špičce.
To je jen jiný pohled na stejnou věc,
jde jen o způsob nakreslení.
Zajímejme se teď o to,
jak spočítat objem tělesa.
Zapomeňme teď na obsahy
jednotlivých obdélníčků.
Představme si jejich rotaci kolem osy x.
Vezměme si pro začátek jeden obdélníček,
který se nám tu tyčí nad dx,

Korean: 
x축에 대해 회전시킵니다
x축에 대해 회전시킵니다
x축에 대해 회전시킵니다
어떻게 되나요?
어떻게 되나요?
동전처럼 생긴 입체가 되는군요
원판 모양으로 말이죠
여기 오른쪽 그림에서도
그려보도록 하겠습니다
그려보도록 하겠습니다
높이가 dx가 되겠네요
이 원판의 부피는
어떻게 구하면 될까요?
그림 밖에도 다시 그려보겠습니다
이 도형을 잘 시각화하는 것은
정말 중요합니다
이게 x축이라면
원판은 이렇게 생겼겠네요
x축이 원판의 가운데를
뚫고 나오게 그려줍니다
이게 원판의 밑면이 되고
높이는 dx입니다

Portuguese: 
Se rotacionar isto em torno do 
eixo x
--Estou tentando fazer o meu melhor--
--rotacionar em torno
do eixo x--
com o que ficamos?
Você terá algo parecido com
uma moeda,
um disco ou um Real.
Deixe-me desenhar isto aqui.
Este mesmo disco irá parecer
com isto aqui.
Com uma espessura dx.
Como podemos calcular o
volume deste disco?
Deixe-me desenhar isto
aqui também.
--É muito importante
visualizar estas formas direito--
Este é o meu eixo x.
Meu disco ficará
deste jeito.
Minha melhor tentativa para
o eixo x é esta aqui.
De fora para o centro,
e isto será a superfície
do meu disco,
e isto bem aqui será
minha espessura dx.

Bulgarian: 
Ако завъртим това около оста х...
Много се старая...
въртим го около оста х,
Какво ще получим?
Получаваме нещо, което
изглежда като монета,
като диск, като някаква монета.
Ще го нарисувам.
Това е някакъв диск,
който изглежда така.
И има дебелина dх.
Как да намерим обема
на този диск?
Ще го пречертая тук.
Много е важно да си 
представим нещата правилно.
Това е оста х.
Нашият диск изглежда ето така.
Оста х минава ето така.
Излиза от центъра.
Това е повърхността
на диска.
Това тук е дебелината dх.

Thai: 
ถ้าคุณหมุนรูปนี้รอบแกน x --
ผมพยายามทำ -- รอบแกน x
คุณหมุนมัน
คุณจะได้อะไร?
 
คุณจะได้รูปที่เป็นเหมือนเหรียญ
เหมือนจาน เหมือนเหรียญอะไรสักอย่าง
ขอผมวาดมันข้างนอกนะ
จานเดิมออกมา จะเป็นแบบนี้
 
และมันลึก dx
เราจะหาปริมาตรของจานนั้นได้อย่างไร?
ขอผมวาดมันใหม่ตรงนี้ด้วย
การมองภาพอย่างถูกต้องนั้นสำคัญมาก
นี่คือแกน x
จานของผมเป็นแบบนี้
ผมพยายามวาดแกน x 
ให้อยู่ตรงนั้นอย่างดีที่สุดแล้ว
มันออกมาจากศูนย์กลาง
แล้วนี่คือพื้นผิวของจานผม
แล้วอันนี้ตรงนี้คือความลึก dx

Czech: 
a nechme ho rotovat kolem osy x.
Co vznikne?
Vznikne cosi, co připomíná minci,
úzký kotouč nebo nízký válec.
Nakreslím ho i vedle do našeho klobouku.
Má výšku ‚dx‘.
Jak teď zjistíme objem takového válce?
Nakreslím ho ještě zvlášť.
Dělat si náčrtky je tu velice přínosné.
Mám osu x.
Můj kotouč, válec, vypadá takto,
osa x prochází jeho středem, kolmo na něj.

English: 
So if you were rotate this
thing around the x-axis--
so I'm trying my best
to-- around the x-axis,
you rotate it.
What do you end up with?
Well, you get something that
looks kind of like a coin,
like a disk, like a
quarter of some kind.
And let me draw it out here.
So that same disk out here
would look something like this.
And it has a depth of dx.
So how can we find the
volume of that disk?
Let me redraw it out here, too.
It's really important to
visualize this stuff properly.
So this is my x-axis.
My disk looks
something like this.
My best attempt at the x-axis
sits it right over there.
It comes out of the center.
And then this is the
surface of my disk.
And then this right over
here is my depth dx.

Korean: 
잘 그려진 것 같네요
명암 처리도 해보겠습니다
명암 처리도 해보겠습니다
이 도형의 부피는
어떻게 구할까요?
다른 원기둥의 부피를 구할 때와 같이
이 원기둥의 밑면의 면적을 구한 후에
높이를 곱해주면 됩니다
이 면의 넓이는 얼마죠?
이미 다 알듯이 원의 넓이는
πr²입니다
그러므로 이 밑면의
반지름을 안다면
밑면의 넓이를 구할 수 있겠네요
그럼 반지름의 길이는 얼마입니까?
반지름의 길이는 원래 직사각형의
높이와 같습니다
그리고 모든 x에 대해 그 값은
f(x)의 값과 같지요
이 예시에서 f(x)는 x²과 같습니다
따라서 이 원판의 반지름은 x⁴이네요
따라서 이 원판의 반지름은 x⁴이네요
따라서 어떠한 x에 대해서
밑면의 넓이는 π(f(x))²이 됩니다
이 경우 f(x)=x²이고요

Thai: 
มันดูดีทีเดียว
แล้วขอผมแรเงามันหน่อย
คุณจะได้เห็นความลึก
แล้วเราจะหาปริมาตรของรูปนี้ได้อย่างไร?
ก็เหมือนจานหรือทรงกระบอกใดๆ
คุณก็แค่ต้องคิดถึงพื้นที่ของหน้านี้
แล้วคูณมันด้วยความลึก
พื้นที่ของฐานนี้เป็นเท่าใด?
เรารู้ว่าพื้นที่วงกลม
เท่ากับพาย r กำลังสอง
ถ้าเรารู้รัศมีของหน้านี้
เราก็หาพื้นที่ของหน้าได้
รัศมีเป็นเท่าใด?
รัศมีก็แค่ความสูง
ของสี่เหลี่ยมมุมฉากเดิม
สำหรับ x ใดๆ ความสูงตรงนี้
จะเท่ากับ f ของ x
และในกรณีนี้ f ของ x คือ x กำลังสอง
ตรงนี้ รัศมีของเราเท่ากับ x กำลังสอง
 
พื้นที่หน้าสำหรับ x ค่าหนึ่ง
จะเท่ากับพายคูณ f ของ x กำลังสอง
ในกรณีนี้ f ของ x คือ x กำลังสอง

Czech: 
Zde je povrch jedné podstavy
a tady výška válce.
A ještě trochu stínování pro efekt.
Co tedy onen objem?
Jako u každého válce nám
bude stačit znát obsah podstavy
a ten vynásobit výškou válce.
Jaký obsah má tedy podstava?
Víme, že obsah kruhu je roven
π krát 'poloměr na druhou'.
Kdybychom tedy znali poloměr,
znali bychom i obsah podstavy.
Jaký má tedy poloměr?
Stejný jako výška obdélníčku,
jež jsme nechali rotovat.
Pro každé ‚x‘, je výška obdélníčku nad ‚x‘
rovna hodnotě funkce v daném ‚x‘.
V tomto případě je
f(x) rovno 'x na druhou'.
Takový je tedy i onen poloměr,
je roven 'x na druhou'.
Obsah podstavy pro dané ‚x‘ je tedy
roven π krát 'f(x) na druhou'.

Bulgarian: 
Това стана много добре.
Само ще оцветя малко,
за да му придам 
повече дълбочина.
Как да намерим обема
на това?
Като всеки диск или
цилиндър,
трябва да намерим
площта на това лице
и после да го умножим
по дебелината.
Колко е площта на основата?
Лицето на кръг е равно
на π по квадрата на радиуса.
Ако знаем радиуса на 
тази основа,
можем да намерим лицето ѝ.
Колко е този радиус?
Радиусът е просто
височината
на първоначалния правоъгълник.
За всяка стойност на х ето тук,
той ще бъде равен на f(х).
В този случай f(х) е х^2.
Значи радиусът е
равен на х^2.
Площта на основата
за дадена стойност на х
ще бъде равна на π по (f(х^2))^2.

English: 
So that looks pretty good.
And then let me just
shade it in a little bit
to give you a little
bit of the depth.
So how can we find
the volume of this?
Well, like any disk
or cylinder, you just
have to think about what
the area of this face
is and then multiply
it times the depth.
So what's the area of this base?
Well, we know that
the area of a circle
is equal to pi r squared.
So if we know the
radius of this face,
we can figure out
the area of the face.
Well, what's the radius?
Well, the radius
is just the height
of that original rectangle.
And for any x, the
height over here
is going to be equal to f of x.
And in this case, f
of x is x squared.
So over here, our radius
is equal to x squared.
So the area of the
face for a particular x
is going to be equal to
pi times f of x squared.
In this case, f
of x is x squared.

Portuguese: 
Ficou muito bom,
mas deixe-me sombreá-la
por dentro
para te dar uma maior
sensação de profundidade.
Como podemos encontrar
volume disto?
Bem, como qualquer disco
ou cilindro,
devemos pensar em termos
da área desta face
multiplicada pela
espessura.
Então qual será a área desta base?
Sabemos que a área do círculo
é igual a pi vezes r ao quadrado.
Então, sabendo o raio desta face,
podemos descobrir sua área.
Mas qual é o seu raio?
O raio é somente a altura
deste retângulo original,
e para qualquer x, a altura
bem aqui será
igual a f de x,
o que neste caso é
x ao quadrado.
Então aqui nosso raio será
x ao quadrado
-- raio igual a x ao quadrado--
E a área desta face, 
para um dado x
será igual a pi vezes
f de x ao quadrado,
neste caso, f de x é
x ao quadrado.

Thai: 
 
แล้วปริมาตรของเราจะเป็นเท่าใด?
ปริมาตรของเราจะเท่ากับ
พื้นที่คูณความลึกตรงนี้
มันจะเท่ากับค่านั้นคูณความลึก คูณ dx
 
ปริมาตรของรูปนี่ตรงนี้ -- ปริมาตร
ของเหรียญนี้ จะเรียกอย่างนั้นก็ได้
จะเท่ากับ
ปริมาตรของผมจะเท่ากับพื้นที่ของผมคูณ
dx ซึ่งเท่ากับพายคูณ x กำลังสอง กำลังสอง
มันจึงเท่ากับพาย -- x กำลังสอง กำลังสอง
ได้ x กำลังสี่ -- พาย x กำลัง 4 dx
 
ทีนี้ พจน์นี่ตรงนี้
มันจะให้ปริมาตรแค่จานหนึ่งในนี้
แต่สิ่งที่เราต้องการคือปริมาตรของหมวกทั้งใบ
หรือแตรหรือกรวยทั้งหมดนี้

Portuguese: 
Agora, como ficará o 
nosso volume?
O volume será nossa área
vezes esta espessura aqui.
Será aquilo vezes a 
espessura, vezes dx
O volume disto aqui
-- do volume
desta moeda, como acho
que você chamará--
será igual a,
meu volume será igual à
minha área vezes
dx, que será igual a pi
vezes x ao quadrado, ao quadrado.
Isto será igual a pi
--x ao quadrado, ao quadrado
é x a quarta--
pi elevado à quarta vezes dx.
Agora, esta expressão aqui,
nos dá o volume de só um
destes discos,
mas o que queremos é o
volume disto tudo,
desta corneta ou cone
inteiro,

English: 
Now, what's our
volume going to be?
Well, our volume is going to be
our area times the depth here.
It's going to be that
times the depth, times dx.
So the volume of this thing
right over here-- so the volume
just of this coin, I
guess you could call it,
is going to be equal to.
So my volume is going to
be equal to my area times
dx, which is equal to pi
times x squared squared.
So it's equal to pi--
x squared squared
is x to the fourth--
pi x to the fourth dx.
Now, this expression
right over here,
this gave us the volume
just of one of those disks.
But what we want is the
volume of this entire hat,
or this entire bugle
or cone-looking,

Bulgarian: 
Тук f(х) е равно на х^2.
А колко е обемът?
Обемът е равен на площта
по дебелината.
Ще бъде равен на това
по дебелината, т.е. по dх.
Значи обемът на това нещо
ето тук е...
само на тази "монета", ако
можем да кажем така,
ще бъде равно на...
Обемът ще е равен на
площта по dх,
което е равно на π(х^2)^2.
Това е равно на –
х^2 на квадрат
е х на четвърта степен,
значи πх^4dх.
Този израз  тук,
това е обемът само
на един от тези дискове.
Но ние търсим обема
на цялата тази "шапка",
на цялата тази фуния,

Czech: 
V tomto případě je
f(x) rovno 'x na druhou'.
Jak bude vypadat náš objem?
Bude to obsah podstavy krát výška.
Výška válečku je ‚dx‘.
Objem našeho válečku,
našeho kotouče nebo mince, jak chcete,
se rovná obsah podstavy krát dx.
To se rovná π krát 'x na druhou',
to celé na druhou,
tedy π krát 'x na čtvrtou' krát dx.
Tento výraz nám dává
objem jednoho kotouče.

Korean: 
이 경우 f(x)=x²이고요
그렇다면 부피는 어떻게 됩니까?
부피는 밑면의 넓이에
높이를 곱하면 되므로
dx를 곱해주면 됩니다
dx를 곱해주면 됩니다
그러므로 여기 있는
동전 모양의 도형의
부피를 계산하면
밑면의 넓이와 높이의 곱과 같으므로
π(x²)²dx가 됩니다
다르게 표현하면
πx⁴dx가 되겠죠
πx⁴dx가 되겠죠
한 개의 원판의
부피일 뿐입니다
우리가 구하고자 하는 것은
이 뿔 모양의 도형의

Czech: 
My chceme přece celý klobouk, celý Ještěd,
celý konec trumpety, dalo by se říct.
Jak na to?
Aplikujeme stejnou myšlenku.
Co když pro objem
všechny ty kotoučky sečteme,
podobně jako jsme sčítali
obdélníčky kvůli obsahu?
Jenom přepnu na psaní jednou barvou.
Sečtěme všechny kotoučky závislé na ‚x‘
pro všechna x od 0 do 2.
To jsou meze,
které jsme si určili už na začátku,
mohli jsme si zvolit i jiné,
mohli jsme vzít jakékoliv dvě hodnoty.
Teď máme 0 a 2.
Proveďme tedy součet
objemů všech těchto mincí.
V limitě, to je, když se šířka mincí
bude pořád změnšovat
a my budeme mít víc a víc mincí,
až jich bude nekonečně mnoho
a budou nekonečně tenké,
dostaneme objem klobouku,
či jak tomu chcete říkat.
Po vypočtení tohoto integrálu
tedy získáme objem.
Umíme ho vypočítat?
Ano, je to standardní integrál,
jaký už známe.

Korean: 
전체 부피입니다
전체 부피입니다
어떻게 계산하면 될까요?
이전과 같은 방법을 적용합니다
이 원판의 부피를
모두 더하는 것이지요
한번 해보겠습니다
색은 한 가지로 쓰겠습니다
πx⁴dx을 모두 더할 겁니다
x의 범위는 0부터 2까지입니다
아까 시작할 때 정한 범위였죠
그냥 임의의 값을 정한 겁니다
0과 2가 아니더라도
어떤 x에 대해서도 가능합니다
여기서는 0부터 2까지로 하겠습니다
이 범위에서 모든
동전 모양의 도형의
넓이를 더해줍니다
이때 원판의 넓이를 점점 줄이면
동전의 개수는 점점 많아질 것이므로
극한을 취하면
우리가 얻고자 했던 뿔 모양의
도형의 부피를 구할 수 있게 됩니다
그러므로 이 정적분을 계산하면
구하고자 했던 부피가 되는 것이지요
한번 해보도록 하겠습니다
이제부터는 그냥 정적분입니다
영상을 보기 전에 먼저

English: 
or I guess you could say the
front-of-a-trumpet-looking
thing.
So how could we do that?
Well, the exact same technique.
What happens if we were to take
the sum of all of these things?
So let's do that, take the
sum of all of these things.
And I'll switch to one color--
pi times x to the fourth dx.
We're going take the sum
of all of these things
from x is equal to 0 to 2.
Those are the boundaries
that we started off with.
I just defined them arbitrarily.
We could do this, really,
for any two x values--
between x is equal
to 0 and x equals 2.
And we're going to take the
sum of the volumes of all
of these coins.
But the limit-- as the depths
get smaller and smaller
and smaller and we have
more and more and more
coins, at the limit,
we're actually
going to get the volume
of our cone or our bugle
or whatever we want to call it.
So if we just evaluate
this definite integral,
we have our volume.
So let's see if we can do that.
And now this is just taking
a standard definite integral.
So this is going to be
equal to-- and I encourage

Bulgarian: 
или на цялата тази
фуния на тромпет.
Как ще го намерим?
Чрез същия метод.
Какво ще стане, ако съберем
всички тези неща?
Да го направим, да съберем
всички тези обеми.
Ще продължа с един цвят –
πх^4dх.
Ще съберем всички
тези обеми от х = 0 до х = 2.
Това са границите,
с които започнахме.
Определих ги произволно.
Можем да вземем
всеки две стойности на х
между х = 0 и х = 2.
И ще съберем обемите
на всички тези "монети".
Но границата... когато
дебелината става все по-малка,
и дисковете стават
все повече и повече,
в границата, всъщност
ще получим обема на
нашия конус или фуния,
или както искаш го наречи.
Така че просто изчисляваме
определен интеграл
и получаваме обема.
Да видим дали
можем да го направим.
Просто взимаме
определен интеграл.
Това ще бъде равно –
насърчавам те

Portuguese: 
ou --qualquer coisa
como um trompete--
Como podemos fazer isto?
Usando exatamente
a mesma técnica.
O que acontece se somarmos
todas estas coisas?
Vamos fazer isto, somar
todas estas coisas.
--Vou trocar a cor--
pi vezes x elevado à quarta dx,
Vamos somar todas estas coisas
de x igual a zero até dois.
Estes são os limites em que
começamos,
foram definidos
arbitrariamente.
Poderíamos ter feito isto
para quaisquer dois valores de x--
entre x igual a zero e
x igual a dois,
e estaríamos somando
os volumes de todas
estas moedas.
No limite, estas espessuras
ficam cada vez menores
e temos mais e mais
moedas que, no limite,
resultarão no volume do nosso cone
ou corneta,
ou o que você queira chamar.
Só calculamos a integral
definida,
e teremos nosso volume.
Vejamos se conseguimos
fazer isto.
Isto somente será o cálculo
de uma integral definida,
o que dará -- te encorajo

Thai: 
จะเรียกว่ารูปทรงเหมือนทรัมเป็ตก็ได้
 
แล้วเราทำได้อย่างไร?
เทคนิคเดิมเลย
เกิดอะไรขึ้นถ้าเราหาผลบวก
ของสิ่งเหล่านี้ทั้งหมด?
ลองทำดู ลองหาผลบวกของรูปท้้งหมดนี้
ผมจะเปลี่ยนเป็นสีเดียวนะ -- 
พายคูณ x กำลัง 4 dx
เราจะหาผลบวกของทั้งหมดนี้
จาก x เท่ากับ 0 ถึง 2
พวกมันคือขอบที่เราเริ่มไว้
ผมเลือกมันมาตามใจ
เราทำอย่างนี้ได้ สำหรับ x สองค่าใดๆ --
ระหว่าง x เท่ากับ 0 กับ x เท่ากับ 2
และเราจะหาผลบวกของปริมาตรเหรียญ
เหล่านี้ทั้งหมด
แต่ลิมิต -- เมื่อความลึกน้อยลง น้อยลง
และน้อยลง เราจะได้เหรียญมากขึ้น มากขึ้นน
และมากขึ้น ที่ลิมิต เราจะ
ได้ปริมาตรกรวยหรือแตร
หรือเราอยากจะเรียกว่าอะไรก็ตาม
ถ้าเราหาค่าอินทิกรัลจำกัดเขตนี้
เราจะได้ปริมาตร
ลองดูว่าเราทำได้ไหม
ตอนนี้ นี่ก็แค่การหาอินทิกรัลจำกัดเขตธรรมดา
อันนี้จะเท่ากับ -- ผมแนะนำ

Korean: 
계산해보시기를 권해드립니다
π를 밖으로 꺼내면
0에서 2까지 ∫x⁴dx의 값에
π를 곱해준 것과 같게 됩니다
색이 조금 마음에 들지 않네요
x⁴의 역도함수는
x^5 / 5입니다
그러므로 π에 x^5 / 5를 곱하고
x에 0과 2를 넣어서
계산해주면 되겠네요
2를 먼저 대입해봅시다
2를 먼저 대입해봅시다
2의 세제곱은 8입니다
2의 네제곱은 16이고요
2의 다섯제곱은 얼마일까요?
2의 다섯제곱에서 0의 다섯제곱을 뺀 값은
2의 다섯제곱은 32이고
0의 다섯제곱은 그냥 0이므로
32/5의 값에
π를 곱해주기만 하면 됩니다
다 되었네요
저 뿔 모양의 부피를 구하는데

Czech: 
Schválně si ho zkuste
nejprve spočítat sami.
π můžeme vytknout.
To dá π krát integrál
od 0 do 2 z 'x na čtvrtou' dx.
Tato barva se mi nelíbí.
Primitivní funkce k 'x na čtvrtou'
je 'x na pátou' lomeno 5.
Máme tedy π krát 'x na pátou' lomeno 5.
Přičemž naše meze jsou 0 a 2.
Dostáváme π krát tento výraz, kde x je 2.
2 na třetí je 8,
2 na čtvrtou je 16,
2 na pátou je…
Napíšu to.
('2 na pátou' lomeno 5)
minus ('0 na pátou' lomeno 5).
To bude rovno…
'2 na pátou' je 32…
Takže to máme π krát 32 děleno 5 minus 0.
32π lomeno 5.
Hotovo, dokázali jsme vypočítat
objem tohoto podivného tělesa.

Thai: 
ให้คุณลองทำก่อนที่ผมจะทำ
เราก็นำพายออกมาได้
มันจะเท่ากับพายคูณอินทิกรัล
จาก 0 ถึง 2 ของ x กำลัง 4 dx
ผมไม่ชอบสีนั้น
ทีนี้ ปฏิยานุพันธ์ของ x กำลัง 4
คือ x กำลัง 5 ส่วน 5
อันนี้จึงเท่ากับพายคูณ x กำลัง 5 ส่วน 5
และเราจะไปจาก 0 ถึง 2
อันนี้จึงเท่ากับพายคูณตัวนี้
หาค่าที่ 2
ลองดู
2 กำลัง 3 ได้ 8
2 กำลัง 4 ได้ 16
2 กำลัง 5 ได้ -- ขอผมเขียนลงไปนะ
2 กำลัง 5 ส่วน 5 ลบ 0 กำลัง 5 ส่วน 5
อันนี้จึงเท่ากับ -- 2
กำลัง 5 ได้ 32 มันจึงเท่ากับพายคูณ
32/5 ลบ -- นี่ก็แค่ 0 --
ลบ 0 ซึ่งเท่ากับ 32 พายส่วน 5
เราก็เสร็จแล้ว
เราสามารถหาปริมาตร

English: 
you to try it out
before I do it.
So we can take the pi out.
So it's going to be equal
to pi times the integral
from 0 to 2 of x
to the fourth dx.
I don't like that color.
Now, the antiderivative
of x to the fourth
is x to the fifth over 5.
So this is going to be equal to
pi times x to the fifth over 5.
And we're going
to go from 0 to 2.
So this is going to be
equal to pi times this thing
evaluated at 2.
Let's see.
2 to the third is 8.
2 to the fourth is 16.
2 to the fifth is-- let
me just write it down.
2 to the fifth over 5 minus
0 to the fifth over 5.
And this is going
to be equal to-- 2
to the fifth is 32, so it's
going to be equal to pi times
32/5 minus-- well,
this is just 0--
minus 0, which is
equal to 32 pi over 5.
And we're done.
We were able to
figure out the volume

Bulgarian: 
да опиташ първо
самостоятелно да го решиш.
Значи можем да изнесем пи.
Това ще бъде равно на
пи по интеграл
от 0 до 2 от х^4dх.
Този цвят не ми харесва.
Примитивната функция на х^4
е х^5 върху 5.
Това е равно на π по х^5/5
Това е от 0 до 2.
Това е равно на
π по това, изчислено за х = 2.
2^3 е равно на 8.
2^4 е 16.
2^5 е – нека да го запиша.
2^5 върху 5 минус 0^5 върху 5.
Това е равно на –
2 на пета степен е 32, така че става
π по 32/5 минус – това е просто 0 –
минус 0, което е 32π/5.
И сме готови.
Успяхме да намерим обема

Portuguese: 
a tentar sozinho antes que
eu faça--
Podemos tirar para fora
o pi,
e ficará igual a pi vezes
a integral
entre zero e dois de x
elevado à quarta dx.
-- Não gostei desta cor--
A antiderivada de x a quarta
é x elevado a cinco
sobre cinco,
e isto será igual a pi vezes
x elevado a cinco sobre cinco,
indo de zero até dois.
Isto será igual a pi vezes
esta coisa
calculada em dois.
Vejamos,
dois ao cubo é oito,
dois à quarta,
dezesseis.
Dois à quinta é --deixe-me escrever
aqui embaixo--
Dois à quinta sobre cinco menos
zero à quinta sobre cinco,
o que dará
-- dois à quinta é
32-- pi vezes
32 sobre cinco menos
--isto será zero--
menos zero, que é igual a
32 pi sobre cinco.
Feito!
Descobrimos o volume
desta forma maluca.

Portuguese: 
Legendado por [Luiz Marangoni]
Revisado por [Soraia Novaes]

Korean: 
성공했습니다
커넥트 번역 봉사단 | 박혜준

English: 
of this kind of wacky shape.

Bulgarian: 
на това странно тяло.

Thai: 
ของรูปประหลาดนี้ได้แล้ว
