Learn about Double Integral 1 Video

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Learn about Double Integral 1

So far we've use integrals to figure out the area under a curve, and let's just review a little bit of the intuition. Although this should hopefully be second nature to you at this point. If it’s not, you might want to review the definite integration videos. But if I have some function, this is the xy plane, that’s the x axis, that’s the y axis. and I have some function, this is y is equal some function of x. give me an x and I’ll give you a y. if I wanted to figure out the area under this curve between x is equal to a and x is equal to b, so this is the area I want to figure out. This area right here. What I do is I split it up into a bunch of columns or a bunch of rectangles. Let me draw one of those rectangles. Where you could view, and there's different ways to do this. But this is just a review. Where you could review, that’s maybe one of the rectangles. So the area of the rectangle is just base times height. Right? We’re going to make these rectangles really skinny, just sum up an infinite number of them. I'm going to make them infinitely small, but let's just call the base of this rectangle dx. And then the height of this rectangle is going to be f of x to that point. If this is x not or whatever, you can just call it f of x, right? That’s the height of that rectangle. And if we wanted to take the sum of all of these rectangles, right? There’s going to be a bunch of them, one there, one there. Then we’ll get the area and we have infinite number of these rectangles. They're infinitely skinny, we have exactly the area under that curve. That’s the intuition behind the definite integral. and the way we write that, it’s the definite integral, were going to take the sums of these rectangles from x is equal to a to x is equal to b. and the sum for the areas that were summing up are going to be the height is f of x and the width is d of x. so it’s going to be f of x times d of x. this is equal to the area under the curve, f of x, y is equal to f of x. from x is equal to a and x is equal to b. and that’s just a little bit of a review. But hopefully, you’ll now see the parallel of how we extend this to taking the volume under a surface. So first of all, what is a surface? Well, if were thinking in 3 dimensions, the surface is going to be a function of x and y. so we can write a surface as, instead of y as a function of x, we can write a surface as z is equal to a function of x and y. so you can kind of view it as the domain, the domain is all of the set of valid things that you can input into a function. Before our domain, at least for most of what we dealt with, was just the x axis. So kind of the real number line in the x direction. Now our domain is the xy plane. We can give any x and any y and we’ll just deal with reals right now and I don’t want to get too technical. And then it’ll pop out another number and if we wanted to graph it, it will be our height. And so that could be a height of a surface. So let me just show you what a surface looks like in case you don’t remember. And we’ll actually figure out the volume under this surface. So this is a surface, I’ll tell you its function in a second. But it’s pretty neat to look at. As you can see, it’s a surface. It’s like a piece of paper that’s bent. Let me rotate it to its traditional form. So this is the x direction, this is the y direction and the height is a function of where we are in the xy plane. So how do we figure out the volume under a surface like this? It seems like a bit of a stretch given what we have learned from this. But what if, and I'm just going to draw an abstract surface here. Let me draw some axis. Let’s say that’s my x axis. That’s my y axis. That’s my z axis. I don’t practice this videos all the time, so I'm often wondering what I'm about to draw. Okay, so that’s x, that’s y and that’s z. and let's say I have some surface. Let's draw something, I don’t know what it is. Some surface. This is our surface, this is a function of x and y. so if you give me a coordinate in the xy plane, say here. I’ll put it into the function and it will give us the z value. I’ll plot it there and it will be a point on the surface. So what we want to figure out is the volume under the surface and we have to specify bounds, right? From here we said x is equal to a to x is equal to b. so let's make a square bound first, because this keeps it a lot simpler. So let's say that the domain or the region, not the domain, the region of the x and y region of this part of the surface under which we want to calculate the volume. Let's say, if the sun was right above the surface, the shadow would be right there. Let me try my best to draw this neatly. So this is what we're going to try to figure out the volume of. So if we wanted to draw in the xy plane, like we just kind of view it as the projection of the surface of the xy plane or the shadow of the surface of the xy plane. What are the bounds? Or you can almost view it, what are the bounds of the domain? Well let's say that this point, that’s 0, 0 in the xy plane. Let's say that this is y is equal to a. that’s this line right here, y is equal to a. and let's say that this line right here is x is equal to b. hope you get that? Right, this is the xy plane. If we have a constant x, there will be a line like that. Our constant y, a line like that. And then we have the area in between it. So how do we figure out the volume under this? Well, if I just want to figure out the area of, let's just say, this sliver. Let's say we had a constant y. let's say I have some sliver, I don’t want to confuse you. Let’s say I have some constant y, I just want to give you the intuition. I don’t know what that is, that’s just an arbitrary y. but for some constant y, what if I could just figure out the area under the curve there, right? How do I figure out just the area under that curve? It will be a function of which y I pick, right? Because if I picked a y here, it will be a different area. If I picked a y there, it will be a different area. But I could view this now as a very similar problem to this one, up here. I could have my dx, let me pick a vibrant color so you can see it. Let’s say that's dx, right? That’s a change in x. and then the height, this height, is going to be a function of the x I have and the y I've picked, right? Although I'm assuming to some degree that’s a constant y. so what would be the area of this sheet of paper, right? It’s kind of a constant y. it’s a sheet of paper within this volume, you can kind of view it. Well, we said the height of each of these rectangles is f of xy, that’s the height. Depends which x and y we picked down here. And then its width is going to be d of x, not d of x, dx. And then if we integrated it from x is equal to 0 which was back here all the way to x is equal to b, what would it look like? It will look like that. x is going from 0 to b. fair enough. And this will actually give us a function of y. this would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. if you give me a y, I can tell you the area of the sliver that corresponds to that y. Now what can I do? If I know the area of any given sliver, what if I multiply the area of that sliver times dy, right? This is a dy, let me do it in a vibrant color. So dy, a very small change in y, right? If I multiply this area times the small dy, then all of a sudden a sliver of volume. Hopefully that makes some sense. Right, I'm making that little cut that I took the area by making it 3 dimensional. So what would be the volume of that sliver? The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of xy dx. That gives us the area of this blue sheet. Now if I multiply this whole thing, times dy. I get this volume, right? I get some depth that this little area that I'm shading right here. It gives depth to that sheet. Now if I added all of those sheets that now have depth, if I took the infinite sum, if I took the integral of this from my lower y bound. From 0 to my upper y bound a. then, at least base on our intuition here, maybe I will have figured out the volume under this surface. But anyway, I didn’t want to confuse you. But that’s the intuition of what were going to do. And I think you're going to find out that actually calculating the volumes are pretty straight forward. Especially when you have fixed x and y bounds. And that’s what we’re going to do in the next video. See you soon.

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So far we've use integrals to figure out the area under a curve, and let's just review a little bit of the intuition. Although this should hopefully be second nature to you at this point. If it’s not, you might want to review the definite integration videos. But if I have some function, this is the xy plane, that’s the x axis, that’s the y axis. and I have some function, this is y is equal some function of x. give me an x and I’ll give you a y. if I wanted to figure out the area under this curve between x is equal to a and x is equal to b, so this is the area I want to figure out. This area right here. What I do is I split it up into a bunch of columns or a bunch of rectangles. Let me draw one of those rectangles. Where you could view, and there's different ways to do this. But this is just a review. Where you could review, that’s maybe one of the rectangles.
So the area of the rectangle is just base times height. Right? We’re going to make these rectangles really skinny, just sum up an infinite number of them. I'm going to make them infinitely small, but let's just call the base of this rectangle dx. And then the height of this rectangle is going to be f of x to that point. If this is x not or whatever, you can just call it f of x, right? That’s the height of that rectangle. And if we wanted to take the sum of all of these rectangles, right? There’s going to be a bunch of them, one there, one there. Then we’ll get the area and we have infinite number of these rectangles. They're infinitely skinny, we have exactly the area under that curve. That’s the intuition behind the definite integral.
and the way we write that, it’s the definite integral, were going to take the sums of these rectangles from x is equal to a to x is equal to b. and the sum for the areas that were summing up are going to be the height is f of x and the width is d of x. so it’s going to be f of x times d of x. this is equal to the area under the curve, f of x, y is equal to f of x. from x is equal to a and x is equal to b. and that’s just a little bit of a review.
But hopefully, you’ll now see the parallel of how we extend this to taking the volume under a surface. So first of all, what is a surface? Well, if were thinking in 3 dimensions, the surface is going to be a function of x and y. so we can write a surface as, instead of y as a function of x, we can write a surface as z is equal to a function of x and y. so you can kind of view it as the domain, the domain is all of the set of valid things that you can input into a function. Before our domain, at least for most of what we dealt with, was just the x axis. So kind of the real number line in the x direction. Now our domain is the xy plane. We can give any x and any y and we’ll just deal with reals right now and I don’t want to get too technical. And then it’ll pop out another number and if we wanted to graph it, it will be our height. And so that could be a height of a surface.
So let me just show you what a surface looks like in case you don’t remember. And we’ll actually figure out the volume under this surface. So this is a surface, I’ll tell you its function in a second. But it’s pretty neat to look at. As you can see, it’s a surface. It’s like a piece of paper that’s bent. Let me rotate it to its traditional form. So this is the x direction, this is the y direction and the height is a function of where we are in the xy plane.
So how do we figure out the volume under a surface like this? It seems like a bit of a stretch given what we have learned from this. But what if, and I'm just going to draw an abstract surface here. Let me draw some axis. Let’s say that’s my x axis. That’s my y axis. That’s my z axis. I don’t practice this videos all the time, so I'm often wondering what I'm about to draw. Okay, so that’s x, that’s y and that’s z. and let's say I have some surface. Let's draw something, I don’t know what it is. Some surface. This is our surface, this is a function of x and y. so if you give me a coordinate in the xy plane, say here. I’ll put it into the function and it will give us the z value. I’ll plot it there and it will be a point on the surface.
So what we want to figure out is the volume under the surface and we have to specify bounds, right? From here we said x is equal to a to x is equal to b. so let's make a square bound first, because this keeps it a lot simpler. So let's say that the domain or the region, not the domain, the region of the x and y region of this part of the surface under which we want to calculate the volume. Let's say, if the sun was right above the surface, the shadow would be right there. Let me try my best to draw this neatly. So this is what we're going to try to figure out the volume of.
So if we wanted to draw in the xy plane, like we just kind of view it as the projection of the surface of the xy plane or the shadow of the surface of the xy plane. What are the bounds? Or you can almost view it, what are the bounds of the domain? Well let's say that this point, that’s 0, 0 in the xy plane. Let's say that this is y is equal to a. that’s this line right here, y is equal to a. and let's say that this line right here is x is equal to b. hope you get that? Right, this is the xy plane. If we have a constant x, there will be a line like that. Our constant y, a line like that. And then we have the area in between it.
So how do we figure out the volume under this? Well, if I just want to figure out the area of, let's just say, this sliver. Let's say we had a constant y. let's say I have some sliver, I don’t want to confuse you. Let’s say I have some constant y, I just want to give you the intuition. I don’t know what that is, that’s just an arbitrary y. but for some constant y, what if I could just figure out the area under the curve there, right? How do I figure out just the area under that curve? It will be a function of which y I pick, right? Because if I picked a y here, it will be a different area. If I picked a y there, it will be a different area.
But I could view this now as a very similar problem to this one, up here. I could have my dx, let me pick a vibrant color so you can see it. Let’s say that's dx, right? That’s a change in x. and then the height, this height, is going to be a function of the x I have and the y I've picked, right? Although I'm assuming to some degree that’s a constant y. so what would be the area of this sheet of paper, right? It’s kind of a constant y. it’s a sheet of paper within this volume, you can kind of view it. Well, we said the height of each of these rectangles is f of xy, that’s the height. Depends which x and y we picked down here. And then its width is going to be d of x, not d of x, dx. And then if we integrated it from x is equal to 0 which was back here all the way to x is equal to b, what would it look like? It will look like that. x is going from 0 to b. fair enough. And this will actually give us a function of y. this would give us an expression so that if I would know the area of this kind of sliver of the volume for any given value of y. if you give me a y, I can tell you the area of the sliver that corresponds to that y.
Now what can I do? If I know the area of any given sliver, what if I multiply the area of that sliver times dy, right? This is a dy, let me do it in a vibrant color. So dy, a very small change in y, right? If I multiply this area times the small dy, then all of a sudden a sliver of volume. Hopefully that makes some sense. Right, I'm making that little cut that I took the area by making it 3 dimensional.
So what would be the volume of that sliver? The volume of that sliver will be this function of y times dy or this whole thing times dy. So it would be the integral from 0 to b of f of xy dx. That gives us the area of this blue sheet. Now if I multiply this whole thing, times dy. I get this volume, right? I get some depth that this little area that I'm shading right here. It gives depth to that sheet. Now if I added all of those sheets that now have depth, if I took the infinite sum, if I took the integral of this from my lower y bound. From 0 to my upper y bound a. then, at least base on our intuition here, maybe I will have figured out the volume under this surface.
But anyway, I didn’t want to confuse you. But that’s the intuition of what were going to do. And I think you're going to find out that actually calculating the volumes are pretty straight forward. Especially when you have fixed x and y bounds. And that’s what we’re going to do in the next video. See you soon.

Transcription by: Scribe4you Transcription Services