Volumes by Slicing (Calculus)
Summary
TLDREl video trata sobre cómo calcular volúmenes mediante secciones en cálculo integral. El presentador explica cómo construir volúmenes a partir de la suma de áreas de figuras geométricas, como semicírculos, cuadrados y triángulos equiláteros, apiladas sobre una región determinada. Se utilizan integrales para calcular estas áreas y luego obtener el volumen total. Se muestra cómo derivar las fórmulas para las diferentes formas y se ilustra el proceso de simplificación de estas integrales, destacando cómo las áreas de las secciones cambian en función de la posición en la región base.
Takeaways
- 📐 El método de volúmenes por cortes se basa en usar integrales para calcular volúmenes sumando áreas de cortes infinitos.
- 🔵 En el ejemplo, la región está delimitada por el eje X y una semicircunferencia con radio 6, y su área se puede calcular usando integrales.
- ➕ Para encontrar el área entre curvas, se usa la fórmula: integral de la función superior menos la función inferior.
- 🍰 El volumen por cortes se genera al apilar formas (como semicírculos, cuadrados o triángulos) sobre una base rectangular.
- 🔺 El área de cada corte semicircular se calcula como 1/2 * π * r^2, donde el radio es la mitad de la longitud de la base del corte.
- ✏️ Para simplificar, el radio se expresa como √(36 - x^2) dividido por 2, resultando en una fórmula para el área de cada semicírculo.
- 🔢 El volumen total con cortes semicirculares se calcula con la integral de -6 a 6 de la fórmula simplificada.
- 📏 También se puede calcular el volumen con cortes cuadrados, usando la fórmula de área de un cuadrado (lado^2).
- 🔻 En el caso de los triángulos equiláteros, la fórmula de área involucra 1/2 * base * altura, y se deduce la altura usando trigonometría.
- 🔧 Cada tipo de corte genera un volumen diferente, dependiendo de la forma de los cortes, pero el principio subyacente es el mismo: integrar áreas para obtener volumen.
Q & A
¿Qué es el concepto de volúmenes por rebanado?
-El concepto de volúmenes por rebanado implica calcular el volumen de un objeto sumando áreas de secciones transversales (rebanadas) a lo largo de una región. Esto se hace utilizando integrales para acumular áreas y obtener el volumen.
¿Cómo se determina el área entre curvas en este ejemplo?
-Para determinar el área entre curvas, se utiliza la integral del límite inferior al superior de la función superior menos la función inferior. En este caso, la función superior es una semicircunferencia y la función inferior es la línea y = 0.
¿Qué forma tiene la región mencionada en el video?
-La región mencionada en el video es una semicircunferencia con radio 6, delimitada por las coordenadas de x que van desde -6 hasta 6 y la curva superior dada por la ecuación y = √(36 - x^2).
¿Cómo se configura la integral para encontrar el volumen cuando se utilizan rebanadas semicirculares?
-Para calcular el volumen con rebanadas semicirculares, se usa la fórmula del área de un semicírculo, que es (1/2)πr². Luego, se configura la integral desde -6 hasta 6, donde r es √(36 - x²) dividido por 2. El volumen se obtiene integrando la expresión π/8 * (36 - x²).
¿Qué sucede con el radio en el caso de los semicírculos en el video?
-El radio de los semicírculos no es la longitud total de la distancia entre las curvas, sino la mitad de esa longitud. En este caso, el radio es √(36 - x²) dividido entre 2.
¿Cómo cambia el volumen si se usan rebanadas cuadradas en lugar de semicírculos?
-Si se usan rebanadas cuadradas en lugar de semicírculos, el área de cada rebanada es simplemente el lado de cada cuadrado al cuadrado, es decir, (√(36 - x²))². El volumen se calcula integrando esta expresión desde -6 hasta 6, lo que da el volumen de un sólido con rebanadas cuadradas.
¿Cómo se determina el área de cada rebanada cuadrada?
-El área de cada rebanada cuadrada se obtiene elevando al cuadrado la longitud del lado, que es √(36 - x²). Al cuadrar esta expresión, se simplifica a 36 - x², que se integra para obtener el volumen total.
¿Qué sucede si las rebanadas son triángulos equiláteros?
-Cuando las rebanadas son triángulos equiláteros, el área de cada triángulo se calcula usando la fórmula (1/2) * base * altura. La base es √(36 - x²) y la altura se puede calcular usando trigonometría o propiedades de triángulos equiláteros.
¿Cómo se determina la altura de un triángulo equilátero en el ejemplo?
-La altura del triángulo equilátero se puede determinar usando trigonometría. Se sabe que el ángulo en un triángulo equilátero es de 60 grados, por lo que se usa la relación tangente de 60° para obtener la altura en términos de la base (√(36 - x²)/2).
¿Cuál es la integral que describe el volumen con rebanadas de triángulos equiláteros?
-La integral para el volumen con rebanadas de triángulos equiláteros es ∫[−6,6] (√3/4) * (36 − x²) dx, donde (√3/4) proviene de la combinación de las fórmulas de la base y la altura del triángulo equilátero.
Outlines
📏 Introducción a los volúmenes por secciones
Este párrafo introduce el concepto de cálculo de volúmenes por secciones mediante integrales, utilizando un ejemplo con una región delimitada por una semicírculo y el eje x. El área entre curvas se explica brevemente, pero el enfoque principal es la transición de áreas a volúmenes, al construir figuras tridimensionales sobre estas áreas mediante cortes o secciones. En este caso, se menciona que las secciones serán semicirculares.
🔄 Explicación del cálculo de volúmenes usando semicírculos
Se describe cómo las rebanadas semicirculares se colocan sobre las áreas rectangulares, sumando los volúmenes de todas estas para obtener el volumen total. El área de cada semicírculo se calcula usando la fórmula para el área de un círculo, pero se ajusta para semicírculos. El radio de estas secciones es la mitad de la distancia desde la raíz de 36 menos x al eje x. Finalmente, se simplifica la fórmula del volumen, dando lugar a una integral que representa el volumen total de la figura con secciones semicirculares.
🔳 Construcción de volúmenes con secciones cuadradas
Este párrafo introduce un nuevo método de seccionado, ahora utilizando cortes cuadrados en lugar de semicírculos. Las secciones cuadradas se colocan sobre la misma base anterior, pero ahora el área de cada sección es simplemente el cuadrado de la longitud del lado, que es la raíz de 36 menos x cuadrado. Se establece la integral para encontrar el volumen de la figura tridimensional resultante de las secciones cuadradas.
🔺 Cálculo de volúmenes con triángulos equiláteros
Aquí se presenta una nueva forma de calcular volúmenes, utilizando cortes con triángulos equiláteros. El párrafo explica cómo se obtiene la base del triángulo y se dedica gran parte del texto a calcular la altura del triángulo utilizando trigonometría, específicamente la tangente de 60 grados. Después de obtener las expresiones para la base y la altura, se formula el área de cada triángulo equilátero y se incorpora en la integral para hallar el volumen total.
Mindmap
Keywords
💡Volumen por rebanadas
💡Integral
💡Semicírculo
💡Área de una sección transversal
💡Radio (R)
💡Pi (π)
💡Cuadrado
💡Triángulo equilátero
💡Región delimitada
💡Eje x
Highlights
Introduction to volumes by slicing using integrals, often following the topic of area between curves.
The region is bounded by two curves: the x-axis (y = 0) and a semicircle (y = sqrt(36 - x^2)) with radius 6.
Setup of integral for finding the area between the curves from -6 to 6, noting symmetry allows for simplifying from 0 to 6.
Transition from calculating area to volume, where rectangles become the base of shapes that are stacked to create volume.
First type of slicing discussed: semicircular slices stacked over the region to find volume.
Explanation of the semicircular slice formula: volume as the integral of the area of semicircles, which is 1/2 π r^2, over the given range.
Radius of each semicircular slice is half the distance across the semicircle, derived from sqrt(36 - x^2) / 2.
Simplifying the integral for semicircular slices: using the formula PI/8 (36 - x^2) dx from -6 to 6.
Introduction to another slicing method: square-shaped slices stacked over the same region for a different volume.
Formula for the volume with square slices: integral of (side length)^2, where the side is the distance sqrt(36 - x^2).
Simplified formula for square slices: integral of (36 - x^2) dx from -6 to 6.
Third slicing method introduced: equilateral triangle slices built on top of the region, resulting in a different shape.
Formula for equilateral triangle slices: 1/2 base times height, where the base is sqrt(36 - x^2) and height is derived using trigonometry.
Height for the equilateral triangle slice calculated as sqrt(3)/2 * sqrt(36 - x^2) using trigonometric relationships.
Final formula for the equilateral triangle volume: integral of (sqrt(3)/4) * (36 - x^2) dx from -6 to 6, combining base and height.
Transcripts
everyone Houston math prep here talking
about volumes by slicing which is
something that you might do in calculus
with integrals right after we've talked
about finding area between curves so
here I've got two curves I've got some
region that's bounded by these curves
the bottom is just the x axis y equals
zero and this top is an upper semicircle
here y equals the square root of 36
minus x squared so it's a semicircle it
has radius six my leftmost point here is
at negative 6 and my rightmost point
here is at 6 if we were just going to
find the area of this region so we would
look at doing the integral from A to B
of the top function minus the bottom
function if I draw my rectangle let's
say this way vertically so if I draw it
vertically top function minus the bottom
function would be the root formula minus
is zero minus zero wouldn't change
anything so we would just get root 36
minus x squared DX from A to B would be
from negative six to six we could go
from zero to six and double it due to
the symmetry not going to go into that
too much here this integral itself is
actually a little bit to deal with so
we're not gonna actually work this but I
just want to make sure that we remember
this idea the idea with volume by
slicing is going to be not how to find
area using this rectangle but imagine
laying this region sort of down on the
desk and this rectangle is no longer
just a rectangle that I'm using to fill
the space to find area within an
integral this rectangle is actually
going to be the base for some shape on
top of it and I'm going to build the
same shape over and over and over and
get a volume basically by making a bunch
of slices that are a similar shape so
we're gonna look at how we'll do that
the first type of slice that I'm going
to use I'm going to build semi circular
slices
on top of my semicircular region so
let's take a look at how that will work
so if I look at this here and I think
about all of my rectangles that would
run through the region and I'm stacking
semi-circle slices on top of each of
those rectangles then I would add up all
the areas on top of those rectangles to
give me a volume so where normally we're
doing an in summing an infinite number
of lengths to give us area we are now
summing an infinite number of areas to
give us volume hopefully that makes some
sense there so now looking at what we
have here so we have a bunch of
cross-sections or slices that are semi
circles the way we'll do volume and the
way a lot of books will do this is they
will say well volume will be the
integral from A to B just like before
they'll use something sometimes like a
of X DX and this a of X is really what
you have to determine in each situation
and it is what is the formula for the
area of each slice so if you look at
this one slice that I have kind of
through the middle of the region how do
I find a formula for the area of any
slice no matter where it is okay so if
we look at the base remember that this
curved part was y equals the square root
of 36 minus x squared and this line over
here was y equals 0 and this point way
over here is that x equals negative 6
and this point way over here is at x
equals 6 this one sort of toward the
middle as far as a slice go so it's
giving us quite a bit of area the slice
is down at the end are much smaller
they're not going to provide much area
in terms of volume so each one gives us
a different amount I need to figure out
a formula for a of X so a of X is going
to be the area of each semicircular
slice so this will depend on the shape
this is a semicircle so what's the area
of a semicircle well the area of a
circle is PI R squared so if I want a
half circle or a semi
circle then the area formula should be
1/2 PI R squared that would be half a
circle as the area for each slice so the
trick then is figuring out what do I put
in for R the rest of this no problem I
got to figure out what is R so if I go
over here and I say well radius would
kind of be along this edge here the
problem is this line here is not the
radius of this semicircle it is the
diameter it's all the way across right
so if you can imagine how would we get a
radius well think about like a point
here only half of this line right would
be a radius that would be an R for us so
what is that well it would be half that
distance from the root down to zero in
other words this distance is going to be
the square root of 36 minus x squared
divided by 2 is really what we've got so
that's what we'll go in for our radius
there let's go ahead and put that in our
a of X so our a of X is going to equal
1/2 pi and then we'll have a root of 36
minus x squared over 2 that's a radius
squared we would do some simplifying
here I'm going to kind of do this all in
one shot so if I square that will take
care of the root the root will go away
if I square the two on the bottom that's
going to give me a 4 and then if I have
4 times this 2 that's also on the bottom
that's going to give me an 8 right so I
really have keep the pie I have PI over
2 times 4 which is 8 and then again the
square gets rid of the root and we have
this so our volume for this object with
all these semicircular slices would be
the integral from negative 6 to 6 PI
over 8 36 minus x squared DX and this
would just be power rules you could bump
the PI over 8 out we're not going to
work out the full integral in this one
we're just going to set it up and give
you the idea for slicing here
okay let's look at another one this time
we're going to do a volume but we're
going to do it with a different type of
slice we're going to do square shaped
slices so same base right but the volume
that we're building on top of this
region is going to have all of its
slices as a square shape so you can see
the volume takes on a very different
tone especially as you look at sort of
the top contours of the shape there so
let's look at how to build volume with
this one so if we have square slices
right so our volume again is going to
equal the integral from A to B formula
for the area of each slice integral DX
right okay so what is the area for this
well this one's not so bad right
I need the area of a square well square
is simply going to equal
whatever the side length is squared
right side times side they're the same
so this length here and we already have
a formula for this so this length we
already know from before right
this was square root 36 minus x squared
that's the side of one side length I
need to square that right so side
squared is going to be the square root
of 36 minus x squared squared in other
words 36 minus x squared so very similar
right based on the shape of the base so
our volume for this one would be the
integral from negative six to six of
simply 36 minus x squared DX and again
you can work this out this isn't too bad
just power rule
plugging in you should be able to get
something for this I think but let's go
ahead and do one more similar but a
different shape this one's a little bit
more complicated we're going to actually
build equilateral triangles on top so
you notice we get a nice little slant on
one side of the volume as we build this
looks more rounded on one side but the
other side has a nice slant surface to
it so we've got equilateral triangles
we're gonna build and we're gonna go
ahead and do that again volume is going
to equal I'm gonna give me some room
here integral from A to B a of xdx our
base is the same so we're still going to
be going from negative six to six we
need to figure out what's the formula
for this equilateral triangle area that
we're dealing with so area well area of
this equilateral triangle any triangle I
guess you could say something like
one-half base times height there are
some other ways to do this but let's
just say you go with this this is a
pretty common way that people will say
area for a triangle so base I think we
have down right the base is just going
to be this distance and we already know
that this distance is the square root of
36 minus x-square from all our other
stuff and then the question becomes what
is the height right what is this height
well you can do a lot of things with
this I guess you could do some maybe
some trigonometry if you know this is
equilateral you know that this is 60
degrees so if you wanted to figure out
some things you could do it a bunch of
different ways if you already know stuff
about similar triangles you could just
figure it out that way you could figure
it out using maybe this half of a right
triangle here and you could say
something like well let's see tangent of
60 degrees is equal to the opposite side
over the adjacent side and this would
just be half of it right so it would be
root 36 minus x squared over 2 lots of
things you can do there so if I simplify
tangent of 60 will get root 3 I can go
ahead and multiply the top and bottom by
two if I want and get something like
this instead so we're getting close to
figuring out what H is so if I go ahead
and multiply both sides by the root on
the bottom and divide both sides by two
and don't worry too much about
simplifying at this point in time then I
would get root three over two times this
square root and we can do some different
things with simplifying there but that
would be our H so this one a little bit
more to deal with as far as getting the
a of X right not as nice as a square
obviously but now we can go ahead and
take this and put it in there put our
base in there as well the nice thing is
if we had 1/2 base times height then
that's 1/2 times the root 36 minus x
squared and then the height this really
gives us another root 3 over 2 and gives
us another of the same root so it
actually allows us to get rid of the
root and combine those so if we go ahead
and turn this into a nice formula all in
one we're going to get something like
first of all integral negative 6 to 6
hasn't changed I'll go ahead and combine
this I could put it out front
but I'll just leave it in the integral
for now so I'd have a root three on top
two times two on the bottom would give
me a four there and then the two roots
multiplied together would just give me a
36 minus x squared so we'd integrate all
of that DX from negative six to six okay
hopefully this gives you an idea it's
really the same thing but you're
building area on top of each rectangle
to create volumes or integrating
infinite number of areas to give volume
by slicing alright good luck with this
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