Defining Double Integration with Riemann Sums | Volume under a Surface

Introduction to double integration using Riemann sums to find volume under a surface in multivariable calculus.

Key Takeaways

Double integration extends the idea of area under a curve to volume under a surface.
Volume approximation uses sums of small boxes formed by subdividing the domain.
Errors in approximation reduce as the number of subdivisions increases and their size decreases.
The limit of these sums as subdivision size approaches zero defines the double integral.
Choice of sample points in each subdivision does not affect the final integral value.

What the video covers

The video explains the analogy between finding area under a curve in single variable calculus and volume under a surface in multivariable calculus.
It introduces the concept of approximating volume by summing up small rectangular boxes instead of rectangles.
The process involves subdividing the domain into smaller rectangles and choosing sample points within each to evaluate the function height.
The volume of each box is calculated as the product of the base area (Δx_k * Δy_k) and the height (function value at chosen point).
By increasing the number of subdivisions and making them smaller, the approximation error decreases.
The video formalizes the definition of double integration as the limit of the sum of volumes of these boxes as the partition size approaches zero.
It emphasizes that the choice of sample points inside each rectangle does not affect the limit.
The partitioning is restricted to rectangular regions for simplicity in this explanation.
The video concludes with the formal Riemann sum notation for double integrals and hints at an example with a specific function.
The concept of taking the limit of sums over partitions is presented as the rigorous definition of volume under a surface.

Chapters

What is the main idea behind double integration in this video?

The video explains double integration as the process of approximating the volume under a surface by summing the volumes of many small rectangular boxes formed by subdividing the domain.

How does the video relate single variable integration to double integration?

It draws an analogy between finding the area under a curve using rectangles in single variable calculus and finding volume under a surface using boxes in multivariable calculus.

Why does the choice of sample points within each subdivision not affect the double integral?

Because as the number of subdivisions increases and their size decreases, the difference caused by choosing different sample points becomes negligible, and the limit defining the integral remains the same.

Full Transcript — Download SRT & Markdown

Speaker A

In this view, we wouldn't talk about how do we find the area under the need of a surface. This multivariable calculus problem is completely analogous to a single variable problem that we answered back in first or calculus, which was the area underneath a curve.

Speaker A

area underneath of a curve and the idea in fact when we talked about the area underneath of a curve was that it would take the region we break it up into a bunch of different rectangles and when we broke the region into a bunch of

Speaker A

The idea, in fact, when we talked about the area underneath a curve, was that it would take the region, we break it up into a bunch of different rectangles. When we broke the region into a bunch of different rectangles, the area of each of those rectangles was easy to compute, and then we can add up all the areas of those rectangles, and that would give us an approximation for the area under the curve of a single variable function.

Speaker A

situation in multivariable calculus is exactly the same except instead of the rectangles I'm adding up little boxes and then in both of these two different stories I could get a better and better approximation by taking a larger and

Speaker A

The situation in multivariable calculus is exactly the same, except instead of the rectangles, I'm adding up little boxes. Then, in both of these two different stories, I could get a better and better approximation by taking a larger and larger number of individual subdivisions. For instance, I could have this larger number of rectangles here or a larger number of little boxes, and I could do it once again.

Speaker A

large number of increasingly small little rectangles then I would get a better and better approximation of my curve let me zoom in on one particular little box here notice that at some point in the box this box is higher than

Speaker A

The more I take my domain and divide it up into an increasingly large number of increasingly small little rectangles, then I would get a better and better approximation of my curve. Let me zoom in on one particular little box here. Notice that at some point in the box, this box is higher than the graph of the function, and at other points, the graph of the function is higher.

Speaker A

surface you're adding up a little bit too little but as you increase the number of them as the size of these gets smaller the argument is that these errors by approximating in this way are gonna get smaller and smaller and

Speaker A

So what we have is a little error appearing, namely that as it sticks its way out above the surface, you're adding up too much, and when it's beneath the surface, you're adding up a little bit too little. But as you increase the number of them, as the size of these gets smaller, the argument is that these errors by approximating in this way are going to get smaller and smaller and smaller.

Speaker A

change Delta Y and I'll find that we the Delta Y but actually that there's going to be a large number of these little rectangles subdividing up the domain and so I actually call it Delta X K and Delta Y K didn't open I put a little bit

Speaker A

Let's see how we can define this a bit more formally. I'm going to focus in just on the domain, and I specified some rectangle in the. This rectangle has a width that I will define to be Delta X, and it has a height change Delta Y.

Speaker A

going to plug it into my function but plug one into my function this is a whole little region I I need to choose a specific point that I will plug into my function so let me choose a pair a pair

Speaker A

And I'll find that we the Delta Y, but actually, that there's going to be a large number of these little rectangles subdividing up the domain, and so I actually call it Delta X K and Delta Y K. Didn't open. I put a little bit Kate such little rectangle in the division of my domain.

Speaker A

approximation our choices might matter in the limit as you go towards an infant large number of infinitesimally small number of these little subdivisions where exactly you choose the XK YK inside of the rectangle is not going to actually matter so that's the domain now

Speaker A

And then the next thing I want to do is I need to figure out what the height of the box is above this little rectangle, and to do that, I'm going to plug it into my function. But plug one into my function. This is a whole little region. I need to choose a specific point that I will plug into my function.

Speaker A

point it's gonna give me the height of my particular box and so the volume of this box is going to be well-based times height so it's the product of the Delta X K Delta Y K in the base and then

Speaker A

So let me choose a pair, a pair X K Y K, and the pair X K Y Kerry is just some point anywhere inside of my rectangle. Doesn't matter you, and I can make different choices as to where that point's gonna be.

Speaker A

to take a limit in some sense let's see how we do this more properly so the first thing I'm going to do if you give me a region and by the way for this video the region has to be a rectangular region a region

Speaker A

And well, for our approximation, our choices might matter. In the limit, as you go towards an infinite large number of infinitesimally small number of these little subdivisions, where exactly you choose the XK YK inside of the rectangle is not going to actually matter. So that's the domain now.

Speaker A

of small little rectangles and the area of each of those rectangles is going to be given by the product of the change in X K times the change in Y K so the total little area that I have is this product

Speaker A

If I go back and look at what it looks like for the graph of the function down the base, I have that Delta XK and Delta Y K, and if I take that specified point, then XK YK, if I take F of that, F of that point, it's gonna give me the height of my particular box.

Speaker A

little rectangles then if I want to approximate the volumes this is not exactly the answer but if I'm gonna do an approximation if my partition of that big rectangular region into n different small little rectangles then I will take

Speaker A

And so the volume of this box is going to be well-based times height, so it's the product of the Delta X K, Delta Y K in the base, and then multiplied by the height, which is the function value at X K Y.

Speaker A

the widths are given by the Delta XK Delta YK then comes the real magic of Riemann integration this volume is an approximation but what I'm going to do is I am going to define the volume under this region by talking about a

Speaker A

So the basic idea I'm going to do is I'm going to compute that volume for this one particular box, and I'm gonna add an ax over all of them, and finally, I'm going to take a limit in some sense. Let's see how we do this more properly.

Speaker A

times the Delta XK Delta Y ok and then what is the limit doing but P is just the name I give for this partition where I took this big region and I partitioned it into the sum of a bunch of little

Speaker A

So the first thing I'm going to do, if you give me a region, and by the way, for this video, the region has to be a rectangular region, a region where it's just a change in X and a change in Y. But if you have such a region, then the first thing that I'm going to do is I'm going to break it up.

Speaker A

go to 0 area which means all the other reptile we're gonna go to zero as well and indeed if the area of these rectangles go to zero what we mean is that the number of those rectangles is going to

Speaker A

I'm going to partition it and do a bunch of small little rectangles, and the area of each of those rectangles is going to be given by the product of the change in X K times the change in Y K. So the total little area that I have is this product of these two things.

Speaker A

number of them is going to infinity nevertheless I take that limit that I'm trying to describe I take that limit for this particular sum and that's me my definition of volume let's see an example the first thing I want to do is

Speaker A

Then if I take this rectangular region, which is broken up into a whole bunch of little rectangles, in each rectangle, I'm going to choose something to get a point. So I'm going to choose an X K Y K inside of each of my little rectangles.

Speaker A

the way I represent this is using one interval times another interval that's my notation for talking about a region that bounds the x and the y so X prime minus 2 and 2 and the y 3 minus 2 into

Speaker A

Then if I want to approximate the volumes, this is not exactly the answer, but if I'm gonna do an approximation, if my partition of that big rectangular region into n different small little rectangles, then I will take the sum. That's what this notation means. Sigma is the sum from 1 up to nth, and I'm adding up the sum of all the volumes of these little boxes where the heights are given by the end of XK YK and where the widths are given by the Delta XK Delta YK.

Speaker A

other number but before I think we can compute so what you see in the graphic here is these four different boxes each of different heights and my approximate volume is going to be the sum of the volumes of those four boxes if I focus

Speaker A

Then comes the real magic of Riemann integration. This volume is an approximation, but what I'm going to do is I am going to define the volume under this region by talking about a particular limit. I'm going to define the volume to be equal to a limit as the length of P goes to 0. We'll talk about that of this sum from 1 up to n of all of these little volumes, the F of XK YK times the Delta XK Delta Y.

Speaker A

bottom left corner of these four different boxes so in other words for the a1 the specific choice that I'm gonna put down here is the point minus 2 2 and then for the a2 a3 and the a4 I

Speaker A

And then what is the limit doing? But P is just the name I give for this partition where I took this big region and I partitioned it into the sum of a bunch of little small rectangles. And when I take that the length of P goes to 0, what I just mean by that is that the largest rectangle in that partition, some might be bigger, some might be smaller. I've seen the largest rectangle is going to go to 0 area, which means all the other rectangles are gonna go to zero as well.

Speaker A

because it gave me a bunch of formulas on the screen then what I want to do is I'm going to say my volume is approximately equal to the sum of four things these four different boxes these four different

Speaker A

And indeed, if the area of these rectangles go to zero, what we mean is that the number of those rectangles is going to infinity. So there's many different ways to talk about this limit, which means that gain a better and better cutting up of my region into a finer and finer partition where the maximum size of any individual a K is going to zero, and the number of them is going to infinity.

Speaker A

for any of those four little rectangles is just two so this is nothing but the sum of the function values times four times two squared I'm going to evaluate this function at the four different points so evaluate the function of minus

Speaker A

Nevertheless, I take that limit that I'm trying to describe. I take that limit for this particular sum, and that's me, my definition of volume. Let's see an example. The first thing I want to do is consider this same function that I've been talking about, 9 minus x squared minus y squared on the same region that I plotted here.

Speaker A

multiplied by 2 squared and that's going to give me a value of 80 so this is an approximation to the area under this particular surface if you have a question about this video leave it down in the comments below

Speaker A

So the X is between -2 and 2, and the Y is between minus 2 and 2. The way I represent this is using one interval times another interval. That's my notation for talking about a region that bounds the x and the y. So X prime minus 2 and 2, and the y 3 minus 2 into.

Speaker A

multivariable calculus so you can check out those videos here and we'll do some more math in the next video

Topics:double integrationRiemann sumsmultivariable calculusvolume under surfaceintegration approximationrectangular partitioncalculuslimit of sumsfunction heightintegration domain

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