What does a triple integral represent?
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- Опубликовано: 12 сен 2024
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0:15 // Recap of what the double integral represents
1:22 // The triple integral has two uses (volume and mass)
1:45 // How to use the triple integral to find volume
8:59 // Why the triple integral does more than the double integral
11:19 // How to use the triple integral to find mass, when the volume has variable density
It can be difficult to visualize what a triple integral represents, which is why in this video we’ll be answering the question, “What am I finding when I evaluate a triple integral?”
In order to answer this question, we’ll compare the triple integral to a double integral, so that we understand exactly how to transition from double integrals into triple integrals. Every piece of the double integral, like the integral, the bounds or limits of integration, the function which is the integrand, and the differential (usually dydx) will all translate into a corresponding piece of the triple integral.
The interesting thing about the triple integral is that it can be used in two ways. In contrast, single integrals only find area under the curve and double integrals only find volume under the surface. But triple integrals can be used to 1) find volume, just like the double integral, and to 2) find mass, when the volume of the region we’re interested in has variable density.
In this way, triple integrals let us do more than we were able to do with double integrals. We’re able to add in the extra dimension of variable density inside the volume, and based on that variable density, find the mass of the volume, as opposed to only being able to find volume, which is what we were limited to in the double integral.
If we want to describe double and triple integrals with words, we can say that for the double integral, we’re integrating a multivariable function f(x,y) over the region R which is defined for x on the interval [a,b] and for y on the interval [c,d], using vertical slices of volume, in order to find the total volume under the surface f(x,y) but above the xy-plane.
In contrast, we can say that for the triple integral, we’re integrating a multivariable function for density f(x,y,z) for the volume B which is defined for x on the interval [a,b] and for y on the interval [c,d] and for z on the interval [r,s], by slicing the volume in three direction to get tiny pieces (or boxes) of volume, in order to find the total mass of the volume.
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“What am I finding when I evaluate a triple integral?” Let's find out! :D
Hi Krista, i am very interested in higher dimensional things.I was wondering for a physical interpretation of a 4 dimensional integral.I am aware of the green theorem of double integral=triple integral,and the path cancelling effect of the interior of surface when doing the triple integral.i meet 6n-dimensional integrals in chemical kinetics in a momentum space, that is 6 dimensions for each of the n particles.But the double=triple thing,and then a four dimensional integral.I was wondering,really wondering ,beyond a mechanical of just doing it what these things are representing.And whether there are any more path canceling effects in higher dimensional integrals.I have asked a lot of people, but never have a answer.I wonder if the sites read the comments sometimes.I understand busy...Um...I have 3 years of math,and some 25 years more of thinking on math.
Krista King , the volume of a solid and mass
Hi! Really well explained and beautiful colors! Which method do you use to write this lessons?
@@ZoeSkye654 Hello there! From my humble knowledge of geometry, Integrals of higher dimension, can represent the volume of higher dimensional body. for example n-dimensional sphere or hypersphere. I'm familiar with it from theoretical-only math course and I never got to deal with actual real world implementations of it(I'm the one who like math to stay abstract and never mind what it could represent in our world :). the volume of higher dimentional sphere can be represented by recursive product of integrals which covers all possible angles and Jacobian of it is also looks pretty cool. What is 10 dimensional ball can be I have no idea and can't imagine it, but exploring it may be exiting :) good luck!
@@michaelshevelin2234 quadruple integral represents the space time continuum a.k.a the end of the world.
This was so cool!! I was completely blown away by the idea that density can be interpreted and treated as another dimension. Words cannot explain how glad I am that you made a video on all this. 2 years later and this video is still helping a lot of people. Thank you so much, Krista!!
This is a more lucid description than I recall from any engineering math lecture 30 years ago, excellent work.
Thank you so much, I'm glad you enjoyed it! :)
Magnificent video!!! I really think online education is going to replace traditional in-class education soon cause you can't promise the quality of the teachers in every school.
I think you're right 😭
well this comment aged well
well well well isn't that accurate today
Damn this information is heaven.. It cleared all my basics of integration.. Really good piece of information.. All of it makes so much of sense now to me..
Excellent viewpoint on integration. If calculus-1 students were given this type of "brief comparison" clarifying how specific concepts evolve through C2, C3, DE, etc., reading the textbook chapter and doing the homework comes alive like it's in 3D. If I (or every math student, for that matter) had access to RUclips, Udemy, and other mathematical tutorials when in school during the 70's and 80's, my life would be astoundingly different. That's assuming, of course, one initiates the passion and commitment necessary to in-depth learning of STEM related studies. I do hope that students born with this technology available to them somehow realize (and respect) that not that long ago another generation had the vision to dig real deep (in baby steps, of course) to create this opportunity. And hopefully, they will do the same.
Thank you Krista King, your impact on the world is immeasurable.
The most lucid explanation on triple integrations i have seen.👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏👏
Thankss sis. I have learnt a new concept of representing 4th dimension in cartesian coordinates
Near the end of the video, when you demonstrate the sum of each individual "box": f(x1, y1, z1)dV + f(x2, y2, z2)dV + ... to build toward the Riemann sum using summation notation, the subscript on each variable doesn't necessarily need to increase with each subsequent term. In the triple Riemann sum, you have a different subscript for each variable (i, j, k). And each subscript (i, j, k) goes from 1 to l, m, n independently. So building the Riemann sum term-by-term could be f(x1, y1, z1)dV + f(x2, y1, z1)dV + ... + f(x1, y2, z1)dV + ... + f(x1, y1, z2)dV + ... It's semantics, but it might help see that we're taking, for example, all the boxes in the first row of the y-direction, the first row of the z-direction, then along the x-direction from "spot" 1 to "spot" L (and so on).
I haven't watched the double integral video, but it might be the same idea for that (based on what you have on the screen from when you derived that).
Let me know if I'm off-base.
I love your channel so much. It has gotten me through CALC 1 - 3.
THANK YOU SO MUCH! I've never had so much thrill while learning math, I am having difficulties finding words to express my joy, thank you so much!
Your explanation is 100 times better than any math book. Fantastic!
Krista, thank you so much for the help with triple integrals-- I was having trouble distinguishing between a double and a triple-- not anymore!
Literally the best explanation I’ve heard, God bless you! Thank you!
totally agree
as a matter of fact, the supreme way to explain the triple integral I've ever seen
greatly appreciate.
I'm so glad it helped! :)
I have no idea how I stumbled across this video, but I couldn't figure out why my momentum equation in my aerodynamics class was making me convert double integral into triple integrals and this completely answered my question. Thank you so much !
I'm glad to see your channel keep growing! :) Keep up the good work.
Thank you so much! :D
Amazing explanation, i'd love if more teachers started taking example from educational videos like this. I was never taught this in my class. Thanks a lot.
This isn't what I expected the triple integral to be but it now makes sense. This is lowkey just a single integral but hidden within it is a double integral so It becomes a triple integral. I'm glad I watched this video because if someone asked me I would have assumed a triple integral would be something with a fourth dimension and we couldn't visualize it with x,y,z coordinates but it's just occurred to me that I wasn't necessarily wrong; the fourth dimension is the volume in this case. I completely makes sense when you look at it from a physics standpoint because you can always have a formula that integrates one thing against another but the that thing itself might have been derived through an integral. Now it makes sense when I here Physicists say time can be thought of another dimension. Thanks
Yeah, exactly! So glad you liked the video... I absolutely love when things "click", especially when you're tying math and physics together! :D
Thank you Krista! :) You make explaining such a difficult topic easy to understand. I struggled to conceptualize triple integrals during my calc 3 course, but now it makes more sense.
I'm so glad it helped! Triple integrals definitely aren't easy! :D
Well done! This video gives you a good intuitive grasp of the triple integral, which is the first port of call for the serious mathematician.
Thanks, john, I'm so glad you enjoyed it! :D
Thank you very much!!! With you and khan academy I'm unstoppable!!
This is such a great video. Really clears my question.
Hey Krista. I was wondering if you wanted me to discuss with you some aspects of the "finite element method". I bet you'll love it. It's everything I do involving my career and what I like, but it's a lot of this stuff.
Thank you again Krista! Your explanations are clear and thorough! -Mom of 7, Christina :)
Awww thank you so much Christina! I’m honored to help in any way that I can!! ❤️❤️
oMG! So long time searching for this... thanks!!
Another great video Krista!
Thank you so much, Ryan, I really appreciate it! :D
Question: does the plank length play a role in the number of subdivisions that can be done when using the limits, as there is no such thing as infinitely small? Does the calculus break down at the plank length as well? In order to find a more accurate description of gravity, would we need to quantize the calculus?
I have my semester exam tomorrow and I'm doin them with the concepts you've helped me with....thank you again! 😘😀
Thank you very much ma'am... It helps.😊
You're welcome, Discovery, I'm happy to help! :D
pure gold! thanks Krista
Thank you very much for making it easy to understand double and triple integrals!
You're welcome, Shweta! I'm so glad it helped! :)
Best math teacher on RUclips. No doubt.
Awww, thank you so much!! :D
Excellent presentation in detail and clearness, thanks a lot dear
I find these videos informative and soothing as well.
Your explaination is great
Awesome work bro
Great video, Krista. Thanks for sharing this.
Your explanation is just so good 😄💖💖
Glad you think so, Ananya! :D
I think the premise the video sets out, is plain wrong, and here's why: you are teaching that a triple integral is "more special" somehow than a double or a single integral, in that it "calculates both volume and mass", and that this isn't the case for the double or single integral. The difference between volume and mass, is having 1 as the integrand, vs. a function changing with respect to one or more of the integrated variables.
*However, this is equally valid for double and single integrals, or even integrals of higher multiples n.*
When you plug in 1 as the integrand, you calculate the "n-dimensional volume" (length, area, volume, hyper-volume ...) of the shape described by the bounds.
For example, allow me to use the notation S{a, b} i(v) dv to symbolise the definite integral of i, with v as the variable that's being integrated over, with lower bound a and higher bound b. Then the single integral S{a, b} 1 dx is the 1-dimensional volume of a 1-dimensional shape bound by a and b: this is a line with an intuitive length of b-a. When we solve the integral, we get [x]{a, b}, or ... b-a. Similarly, the double integral S{c, d}S{a, b} 1 dxdy calculates the 2-dimensional volume of a shape bound by a and b on one axis, and c and d on the other: a rectangle with an area of (b-a)*(d-c), intuitively, and when we solve the double integral from the inside out, we get S{c, d}(b-a)dy =(b-a)*d - (b-a)*c = (b-a)*(d-c), the same thing.
When you plug in a variable function as the integrand, this "extends" the n-multiple integral to a dimensionality of n+1. This means you're now describing a property of the domain you're integrating over, like temperature, density, or ... "n+1"-volume.
Saying the double integral and triple integral are both capable of calculating volume, isn't wrong, but it is making a false comparison: a triple integral calculates volume, its form of n-dimensional volume, natively. This means it could be extended to calculate the mass of the volume (by plugging in a density function), because it hasn't been extended to n+1 dimensions yet. The double integral, however, can't calculate both volume and the mass *of that same volume* because it natively calculates area (again, its n-dimensional volume, through the integrand of 1), and as all integrals, can only be extended once.
Can double and single integrals calculate mass? Of course they can. It just depends on how tightly you adhere to the most realistic way of defining "density": if you live in a 2- or 1-dimensional world, there could still be such a thing as density, but just in terms of the available spatial dimensions. (In fact, this is used quite a lot in mechanics, where a sheet of metal may have a density in kg/m², or where a pole may have a density of kg/m. The formula for mass, then, isn't V*rho, but A*rho and L*rho.) This means that, when you calculate the volume over a certain 2D domain using a double integral, it mathematically is doing the same thing as calculating the mass of that 2D domain: the integrand function in the latter specifies *how much mass there is per unit of area* in every point of the 2D domain, while the former specifies *how much volume above the xy-plain there is per unit of area* in every point of the 2D domain. The latter is expressed in kg/m², the former in m³/m², and multiplied by dA (in m²), this gives units of kg and m³, that can be summed to get the mass of the 2D domain, or the volume above that 2D domain.
Notice that when you do this "extending" to n+1 dimensions to calculate n+1-dimensional volume (e.g. using a double integral, natively 2D, to calculate volume, natively a 3D-property), you could also just write this as a new n+1-multiple integral, with the extra integral now having the double integral's integrand as its upper bound and 0 as its lower bound, whilst now having an integrand of 1. The double integral calculating volume has been "natified", and can now be used to calculate a property of the volume it describes, since the integrand is 1 instead of a function.
An n-multiple integral is always able to calculate its native (integrand is 1) n-dimensional volume, and it is also able to calculate the total of a property *of that same n-dimensional volume* (integrand is a spatial function expressed in unit/meter^n), but *one property at a time* only.
Hope this helped some of you confused on why the double integral or single integral couldn't just take on the integrand of 1 and have an analogous definition: of course they can.
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Love You...👌👍..
watching many videos, all of them really help full..!!
keep Going well..☺👍
This does not give geometrical interpretation of the triple integral(s). You can not make assumptions to expect this kind of interpretation i.e. to assume same/different density for various dimensions of cube(s). It is good about use of any multiple integrals: solving them as definite integrals, reducing them to single integral etc, but geometrical interpretation is by far possible by the use of doube integrals. That is by our knowledge of three dimensions, not considering time as dimension because it is still truly unexplained by our knowledge. Density is not a dimension, nor any other physical quantity in that meaning.
Geometrical interpretation of multiple (larger than double) integrals could be something extraordinary.
Thank you mam, searching for a week in the you tube didn't get this excellent explanation. love from 🇮🇳 india
You're welcome, I'm so glad it helped! :)
Thank you Krista for your clear explanation :)
You're welcome, Tal! I'm happy to help! :)
great video, thank you so much!
Hey Krista, i am in grade 12 now and i want to study actuarial science next year. I want to prepare on advance by using your courses on udemy. Which courses should i do to prepare for my firts year?
great explanation!! Thanks!
Thanks, Kevin, I'm so glad it helped! :)
Really nice video & amazing explanation skills.
Will f(x, y, z) always represents density?
As I heard triple integral represents hyper volume
So in the case where we calculate the volume using the triple integral ("1" as the integrand), are we saying that the density at each point is equal to 1? And thus we assume an equal distribution of density?
You could think of it that way. Maybe you are imagining the solid, to be made of a kind of plastic, that matches liquid water's density. Its mass in grams, would therefore equal its volume in cubic centimeters.
Teacher as well as teaching r awesome...!!!
Thank you so much! :D
I had this question. Thankyou for the great video
Great Video. Thank you, Krista.
You are so welcome, Anaekh! :D
Hello ! , i'am glad your back !
so , i really don't know
intergrals are said to be antiderivatives
so if integral means area under the curve , then how is the derivative is related to integral, grapically
and how we knew that integral and antiderivatives are equivalent ?
and thank you ! :D
Integrals and antiderivatives are just two different words for the same thing, so we don't have to think about why they're equivalent, we're just using different words to describe one value. And the derivative is the opposite of the integral, they are operations that undo each other! :)
Absolutely loved it.. Super explanation..👌👌👌
Nice interpretation of triple integral. Really it helped alot to me....thanks.
Just one word: WOW 🔥❤️😭
Awesome video!
great video on triple integrals. i'm having a hard time with D though. i'm having trouble with D where it varies in z for example. but as an aside though, these techniques are in reality, the least used methods in practical applications. they are almost purely academic.
Great job!
*Well explained & nicely illustrated diagrammatically Krista, but can you please tell me is there's a quadrupple and quintupple intregral as well? If yes, It would be great if you do a tutorial on them too.Thank you.*
You can continue to add stages of integration and variables of integration, but it will get harder and harder to contextualize it in an application that you could easily understand, and see its relevance to the real world. Maybe it's for a video game that takes place in 4-dimensional hyperspace.
For a quadruple integral, one example I can think of (even when limited to 3D space) is the following:
A block has a temperature distribution that is not uniform, and a specific heat capacity that significantly varies as a function of temperature. You'd like to know the total amount of thermal energy stored in this block at this temperature condition, relative to a reference state where it is uniformly at 0 Celsius.
This would be a quadruple integral of c(x, y, z, T) dx dy dz dT, where c is the specific heat capacity.
Thanks, exactly what I was looking for.
OMG, thank you so much! this is so beautiful
This is a great explanation!
Thank you so much, Jasper! :)
You are just awesome explainer ...@Krista King..
Thank you! :D
Amazing explanation!!! Thanks........!!
You're welcome, Ratnadip! :)
Excellent 👍
Thank You Mam,
The way you introduced the density Concept into the triple integral problem made me understand what does a triple integral really means,!!
Oh good! I'm so glad it made sense! :D
OMG.. you are perfect
Not gonna lie your voice is soothing. Plus your really smart. Yep I’d date you
Hi Krista!
Can you solve the surface of x²+y²=2y revolved about the x-axis?
Thanks a bunch❤
You're welcome, so glad it helped! :)
Very nice. Thanks!
Nice video indeed, thank you :)
You're welcoem, Sergio, I'm glad you liked it! :)
Out of curiosity, is there a fundamental limit to which division can be made because of the plank length. Is it not accurate to say that the integral divides the box up into infinite increments because there is a fundamental limit to space?
Thank you for the lesson.
Pure mathematics ignores this limitation of reality, and assumes an infinitely divisible continuum of space and time. Even if you had considered the Planck length and other quantized limitations of reality, measurement uncertainty at the human scale (centimeters, meters, kilograms, etc), would dwarf the inaccuracies created by the quantization of space and time.
@@carultch Thank you. Makes sense.
Thank u soooooo much 😘😘😊
You're welcome, Epic! :D
completed the play list wish me luck in the tomorrow's exam.
I hope you crushed it! :)
Hey Krista so what is the difference between finding the volume using double and triple integrals versus using disk and washer method in AP calculus for example? Are they like not the same thing?
Good question. The disk/washer methods from introductory calculus, are special cases of repeated integrals, where symmetry works in our favor to simplify the work. This eliminates integration stages and replaces them with standard multiplication.
As an example, for the disk method for a body of revolution, you are given radius as a function of x-position. Your integrand is pi*r^2 dx where r is a function of x, gives you the volume of the thin disk. "Adding" it up along x gives the total volume. The pi*r^2, is a hidden integral, where you are integrating 1/2*r^2 dtheta, the area of each thin "pizza slice". Since r is constant relative to theta for a body of revolution, this pens out to just be multiplying by the whole spread of theta, which is 2*pi. If we had a more exotic shape than an axisymmetric body of revolution, it would be a double integral to find volume.
So the main purpose then of the triple integral is to find mass? And it's only that different from the double integral in the respect that instead of dividing up a flat domain on the xy plane we are dividing up some 3-dimensional shape?
I wouldn't consider that the main application of triple integrals, it's just the easiest example to introduce and visualize, because it doesn't require other background knowledge from other subjects, beyond what the audience most likely already understands.
There are applications in fluid mechanics that people use in the real world, but that have nothing to do with mass. For instance, finding the total bending load on an airplane wing, from responding to the uplift of air. Also applications in statistics, that have nothing to do with 3D geometry. But this would require a lot of background knowledge, that's beyond the scope of the topic.
Well explained
good teacher
okay... but you defined f(x, y, z) to be a density function. what I was looking forward to understanding was for the meaning of a triple integral as it is, without giving f(x, y, z) a physical interpretation.
It's a continuous total of a function across a field of its group of input values. That's a very vague explanation, but without contextualizing it in an application, it's hard to be any more specific than that.
What about Nth integral?
thank you
Pls explain what surface integrals are!!!
Why would i ever use integrals to find the volume of a box instead of l x w x h?
Because triple integrals also let you find the density of a 3D object, not just the volume, plus often the volume of the object can’t be found with a simple geometry formula if the object is irregular, so triple integrals are still really useful! 💪
Interesting.
It never occurred to me that you could consider a fourth dimension to be a quality of a 3-D object. This opens some interesting possibilities...
interesting stuff !
Thanks! :D
You say we can't find mass using double integral, it's not true, we can easily find mass using double integrals if the function of density is of two variables
Tq. Bro
thanks mam
Eurica all over my mind!!!
Oh good!! That makes me so happy! :)
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Awsome
Thanks, sina! :)
And quadruple integral represents the space time continuum a.k.a the end of the world.
There are applications of quadruple integrals, when limited to our 3D space.
One such example:
You have a body made of a material whose specific heat capacity varies significantly with temperature, and its temperature is not uniform. You'd like to know the total thermal energy of this body, relative to a base case (such as when it is uniformly at zero Celsius). Suppose you know its temperature distribution, and how specific heat depends on temperature.
This ultimately is an integral of c(x, y, z, T) dx dy dz dT, the specific heat capacity, as a function of temperature, and the three dimensions of space.
nice but plz make on algebra
awesome video! Thank you!
Great video!
Thanks Michael!