Sentence stringlengths 131 8.39k | video_title stringlengths 12 104 |
|---|---|
Well, what is the anti-derivative of, we're integrating with respect to z first, so what's the anti-derivative of x, y, z with respect to z? Well, it's, let's see, this is just a constant, so it'll be x, y, z squared over 2. Right? Yeah, that's right. And then we'll evaluate that from 2 to 0. And so you get, I know I'm going to run out of space, so you're going to get 2 squared, which is 4, divided by 2, which is 2, so it's 2xy minus 0. So when you evaluate just this first integral, you get 2xy, and now you have the other two integrals left. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Yeah, that's right. And then we'll evaluate that from 2 to 0. And so you get, I know I'm going to run out of space, so you're going to get 2 squared, which is 4, divided by 2, which is 2, so it's 2xy minus 0. So when you evaluate just this first integral, you get 2xy, and now you have the other two integrals left. So I didn't write the other two integrals down, maybe I'll write it down. So then you're left with two integrals. You're left with dy and dx, and y goes from 0 to 4, and x goes from 0 to 3. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So when you evaluate just this first integral, you get 2xy, and now you have the other two integrals left. So I didn't write the other two integrals down, maybe I'll write it down. So then you're left with two integrals. You're left with dy and dx, and y goes from 0 to 4, and x goes from 0 to 3. I'm definitely going to run out of space. And now you take the anti-derivative of this with respect to y. So what's the anti-derivative of this with respect to y? | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
You're left with dy and dx, and y goes from 0 to 4, and x goes from 0 to 3. I'm definitely going to run out of space. And now you take the anti-derivative of this with respect to y. So what's the anti-derivative of this with respect to y? Let me erase some stuff, just so I don't get too messy. I was given the very good suggestion of making it scroll, but unfortunately I didn't make it scroll enough this time, so I can delete this stuff, I think. Oops, I deleted some of that, but you know what I deleted. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So what's the anti-derivative of this with respect to y? Let me erase some stuff, just so I don't get too messy. I was given the very good suggestion of making it scroll, but unfortunately I didn't make it scroll enough this time, so I can delete this stuff, I think. Oops, I deleted some of that, but you know what I deleted. OK, so let's take the anti-derivative with respect to y, I'll start it up here. We're out of space. OK, so the anti-derivative of 2xy with respect to y is y squared over 2, 2's cancel out, so you get xy squared, and y goes from 0 to 4, and then we still have the outer integral to do. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Oops, I deleted some of that, but you know what I deleted. OK, so let's take the anti-derivative with respect to y, I'll start it up here. We're out of space. OK, so the anti-derivative of 2xy with respect to y is y squared over 2, 2's cancel out, so you get xy squared, and y goes from 0 to 4, and then we still have the outer integral to do. x goes from 0 to 3 dx. And when y is equal to 4, you get 16x, and then when y is 0, the whole thing is 0. So you have 16x integrated from 0 to 3 dx, and that is equal to 8x squared. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
OK, so the anti-derivative of 2xy with respect to y is y squared over 2, 2's cancel out, so you get xy squared, and y goes from 0 to 4, and then we still have the outer integral to do. x goes from 0 to 3 dx. And when y is equal to 4, you get 16x, and then when y is 0, the whole thing is 0. So you have 16x integrated from 0 to 3 dx, and that is equal to 8x squared. And we evaluate it from 0 to 3. When it's 3, 8 times 9 is 72, and 0 times 8 is 0. So the mass of our figure, the volume we figured out last time was 24 meters cubed, I erased it, but if you watched the last video, that's what we learned. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So you have 16x integrated from 0 to 3 dx, and that is equal to 8x squared. And we evaluate it from 0 to 3. When it's 3, 8 times 9 is 72, and 0 times 8 is 0. So the mass of our figure, the volume we figured out last time was 24 meters cubed, I erased it, but if you watched the last video, that's what we learned. But its mass is 72 kilograms. And we did that by integrating this three-variable density function, this function of three variables, or in three dimensions, you can view it as a scalar field, right? At any given point, there is a value, but not really a direction, and that value is a density. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So the mass of our figure, the volume we figured out last time was 24 meters cubed, I erased it, but if you watched the last video, that's what we learned. But its mass is 72 kilograms. And we did that by integrating this three-variable density function, this function of three variables, or in three dimensions, you can view it as a scalar field, right? At any given point, there is a value, but not really a direction, and that value is a density. But we integrated this scalar field in this volume. So that's kind of the new skill we learned with the triple integral. And in the next video, I'll show you how to set up a more complicated triple integrals. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
At any given point, there is a value, but not really a direction, and that value is a density. But we integrated this scalar field in this volume. So that's kind of the new skill we learned with the triple integral. And in the next video, I'll show you how to set up a more complicated triple integrals. But the real difficulty with triple integrals is, and I think you'll see that your calculus teacher will often do this. When you're doing triple integrals, unless you have a very easy figure like this, the evaluation, if you actually wanted to analytically evaluate a triple integral that has more complicated boundaries or more complicated, for example, a density function, the integral gets very hairy very fast, and it's often very difficult or very time consuming to evaluate it analytically, just using your traditional calculus skills. So you'll see that on a lot of calculus exams, when they start doing the triple integral, they just want you to set it up. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And in the next video, I'll show you how to set up a more complicated triple integrals. But the real difficulty with triple integrals is, and I think you'll see that your calculus teacher will often do this. When you're doing triple integrals, unless you have a very easy figure like this, the evaluation, if you actually wanted to analytically evaluate a triple integral that has more complicated boundaries or more complicated, for example, a density function, the integral gets very hairy very fast, and it's often very difficult or very time consuming to evaluate it analytically, just using your traditional calculus skills. So you'll see that on a lot of calculus exams, when they start doing the triple integral, they just want you to set it up. And they take your word for it, that you've done so many integrals so far that you could take the antiderivative. And sometimes, if they really want to give you something more difficult, they'll just say, well, switch the order. This is the integral when we're doing with respect to z, then y, then x. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So you'll see that on a lot of calculus exams, when they start doing the triple integral, they just want you to set it up. And they take your word for it, that you've done so many integrals so far that you could take the antiderivative. And sometimes, if they really want to give you something more difficult, they'll just say, well, switch the order. This is the integral when we're doing with respect to z, then y, then x. We want you to rewrite this integral when you switch the order. And we will do that in the next video. See you soon. | Triple integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So let's actually compute it. And I think it'll all become a lot more concrete. So let's say I have the surface z. And it's a function of x and y. And it equals xy squared. It's a surface in three-dimensional space. And I want to know the volume between this surface and the xy plane, and the domain in the xy plane that I care about is x is greater than or equal to 0 and less than or equal to 2, and y is greater than or equal to 0 and less than or equal to 1. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And it's a function of x and y. And it equals xy squared. It's a surface in three-dimensional space. And I want to know the volume between this surface and the xy plane, and the domain in the xy plane that I care about is x is greater than or equal to 0 and less than or equal to 2, and y is greater than or equal to 0 and less than or equal to 1. And let's see what that looks like, just so we have a good visualization of it. So I graphed it here. And we can rotate around. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And I want to know the volume between this surface and the xy plane, and the domain in the xy plane that I care about is x is greater than or equal to 0 and less than or equal to 2, and y is greater than or equal to 0 and less than or equal to 1. And let's see what that looks like, just so we have a good visualization of it. So I graphed it here. And we can rotate around. This is z equals xy squared. And this is the bounding box, right? x goes from 0 to 2, y goes from 0 to 1. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And we can rotate around. This is z equals xy squared. And this is the bounding box, right? x goes from 0 to 2, y goes from 0 to 1. So we literally want this. You can almost view it the volume, well, not almost, exactly view it as the volume under this surface, between this surface, the top surface, and the xy plane. And I'll rotate it around so you can get a little bit better sense of the actual volume. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
x goes from 0 to 2, y goes from 0 to 1. So we literally want this. You can almost view it the volume, well, not almost, exactly view it as the volume under this surface, between this surface, the top surface, and the xy plane. And I'll rotate it around so you can get a little bit better sense of the actual volume. Let me rotate it. Now I should use the mouse for this. So it's this space underneath here. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And I'll rotate it around so you can get a little bit better sense of the actual volume. Let me rotate it. Now I should use the mouse for this. So it's this space underneath here. It's like a makeshift shelter or something. And I can rotate it a little bit. So you can see whatever's under this, between the two surfaces, that's the volume. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So it's this space underneath here. It's like a makeshift shelter or something. And I can rotate it a little bit. So you can see whatever's under this, between the two surfaces, that's the volume. Whoops, I flipped it. There you go. So that's the volume that we care about. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So you can see whatever's under this, between the two surfaces, that's the volume. Whoops, I flipped it. There you go. So that's the volume that we care about. Let's figure out how to do it. And we'll try to gather a little bit of the intuition as we go along. So I'm going to draw a not as impressive version of that graph. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So that's the volume that we care about. Let's figure out how to do it. And we'll try to gather a little bit of the intuition as we go along. So I'm going to draw a not as impressive version of that graph. But I think it'll do the job for now. Let me draw my axes. That's my x-axis. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So I'm going to draw a not as impressive version of that graph. But I think it'll do the job for now. Let me draw my axes. That's my x-axis. That's my y-axis. That's my z-axis. x, y, z. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
That's my x-axis. That's my y-axis. That's my z-axis. x, y, z. And x is going from 0 to 2. Let's say that's 2. y is going from 0 to 1. And so we're taking the volume above this rectangle, the xy plane, and then the surface, I'm going to try my best to draw it. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
x, y, z. And x is going from 0 to 2. Let's say that's 2. y is going from 0 to 1. And so we're taking the volume above this rectangle, the xy plane, and then the surface, I'm going to try my best to draw it. I'll draw it in a different color. I'm looking at the picture. At this end, it looks something like this. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And so we're taking the volume above this rectangle, the xy plane, and then the surface, I'm going to try my best to draw it. I'll draw it in a different color. I'm looking at the picture. At this end, it looks something like this. And then it has a straight line. Let's see if I can draw this surface going down like that. And then if I was really good, I could shade it. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
At this end, it looks something like this. And then it has a straight line. Let's see if I can draw this surface going down like that. And then if I was really good, I could shade it. It looks something like this. This surface looks something like that. And this right here is above this. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then if I was really good, I could shade it. It looks something like this. This surface looks something like that. And this right here is above this. This is the bottom left corner. You can almost view it. So let me just say the yellow is the top of the surface. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And this right here is above this. This is the bottom left corner. You can almost view it. So let me just say the yellow is the top of the surface. And then this is under the surface. So we care about this volume underneath here. Let me show you what the actual surface is. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So let me just say the yellow is the top of the surface. And then this is under the surface. So we care about this volume underneath here. Let me show you what the actual surface is. So this volume underneath here. I think you get the idea. So how do we do that? | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let me show you what the actual surface is. So this volume underneath here. I think you get the idea. So how do we do that? Well, in the last example, we said, well, let's pick an arbitrary y. And for that y, let's figure out the area under the curve. So if we fix some y, when you actually do the problem, you don't have to think about it in this much detail. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So how do we do that? Well, in the last example, we said, well, let's pick an arbitrary y. And for that y, let's figure out the area under the curve. So if we fix some y, when you actually do the problem, you don't have to think about it in this much detail. But I want to give you the intuition. So if we pick just an arbitrary y here. So on that y, we could think of it, if we have a fixed y, then the function of x and y, you can almost view it as a function of just x for that given y. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So if we fix some y, when you actually do the problem, you don't have to think about it in this much detail. But I want to give you the intuition. So if we pick just an arbitrary y here. So on that y, we could think of it, if we have a fixed y, then the function of x and y, you can almost view it as a function of just x for that given y. And so we're kind of figuring out the value of this, of the area under this curve. So this is kind of an up-down curve for a given y. So if we know a y, we can figure out then, for example, if y was 5, this function would become z equals 25x. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So on that y, we could think of it, if we have a fixed y, then the function of x and y, you can almost view it as a function of just x for that given y. And so we're kind of figuring out the value of this, of the area under this curve. So this is kind of an up-down curve for a given y. So if we know a y, we can figure out then, for example, if y was 5, this function would become z equals 25x. And then that's easy to figure out the value of the curve under. So we'll make the value under the curve as a function of y. We'll pretend like it's just a constant. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So if we know a y, we can figure out then, for example, if y was 5, this function would become z equals 25x. And then that's easy to figure out the value of the curve under. So we'll make the value under the curve as a function of y. We'll pretend like it's just a constant. So let's do that. So if we have a dx, that's our change in x. And then our height of each of our rectangles is going to be a function, is going to be z. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We'll pretend like it's just a constant. So let's do that. So if we have a dx, that's our change in x. And then our height of each of our rectangles is going to be a function, is going to be z. The height is z, which is a function of x and y. So we can take the integral. So the area of each of these is going to be our function, xy squared, I'll do it here because I'll run out of space, xy squared times the width, which is dx. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then our height of each of our rectangles is going to be a function, is going to be z. The height is z, which is a function of x and y. So we can take the integral. So the area of each of these is going to be our function, xy squared, I'll do it here because I'll run out of space, xy squared times the width, which is dx. If we want the area of this slice for a given y, we just integrate along the x-axis. We're going to integrate from x is equal to 0 to x is equal to 2. From x is equal to 0 to 2. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So the area of each of these is going to be our function, xy squared, I'll do it here because I'll run out of space, xy squared times the width, which is dx. If we want the area of this slice for a given y, we just integrate along the x-axis. We're going to integrate from x is equal to 0 to x is equal to 2. From x is equal to 0 to 2. Fair enough. Now, we just don't want to figure out the area under the curve at one slice for one particular y. We want to figure out the entire area of the curve. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
From x is equal to 0 to 2. Fair enough. Now, we just don't want to figure out the area under the curve at one slice for one particular y. We want to figure out the entire area of the curve. So what we do is, we say, OK, fine. The area under the curve, not the surface, under this curve for a particular y is this expression. Well, what if I gave it a little bit of depth? | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We want to figure out the entire area of the curve. So what we do is, we say, OK, fine. The area under the curve, not the surface, under this curve for a particular y is this expression. Well, what if I gave it a little bit of depth? If I multiplied this area times dy, then it would give me a little bit of depth. We'd kind of have a three-dimensional slice of the volume that we care about. I know it's hard to imagine. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Well, what if I gave it a little bit of depth? If I multiplied this area times dy, then it would give me a little bit of depth. We'd kind of have a three-dimensional slice of the volume that we care about. I know it's hard to imagine. Let me bring that here. So if I had a slice here, this is what we just figured out, the area of a slice. And then I'm multiplying it by dy to give it a little bit of depth. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
I know it's hard to imagine. Let me bring that here. So if I had a slice here, this is what we just figured out, the area of a slice. And then I'm multiplying it by dy to give it a little bit of depth. So you multiply it by dy to give it a little bit of depth. And then if we want the entire volume under the curve, we add all the dy's together, take the infinite sum of these infinitely small volumes, really, now. And so we will integrate from y is equal to 0 to y is equal to 1. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then I'm multiplying it by dy to give it a little bit of depth. So you multiply it by dy to give it a little bit of depth. And then if we want the entire volume under the curve, we add all the dy's together, take the infinite sum of these infinitely small volumes, really, now. And so we will integrate from y is equal to 0 to y is equal to 1. I know this graph is a little hard to understand, but you might want to re-watch the first video. I had a slightly easier to understand surface. So now how do we evaluate this? | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And so we will integrate from y is equal to 0 to y is equal to 1. I know this graph is a little hard to understand, but you might want to re-watch the first video. I had a slightly easier to understand surface. So now how do we evaluate this? Well, like we said, you evaluate from the inside and go outward. And it's almost like taking a partial derivative in reverse. So we're integrating here with respect to x. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So now how do we evaluate this? Well, like we said, you evaluate from the inside and go outward. And it's almost like taking a partial derivative in reverse. So we're integrating here with respect to x. So we can treat y just like a constant, like it's like the number 5 or something like that. So it really doesn't change the integral. So what's the antiderivative of xy squared? | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So we're integrating here with respect to x. So we can treat y just like a constant, like it's like the number 5 or something like that. So it really doesn't change the integral. So what's the antiderivative of xy squared? Well, the antiderivative of xy squared, I want to make sure I'm color consistent. Well, the antiderivative of x is x to the 1 half, sorry, x squared over 2. And then y squared is just a constant, right? | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So what's the antiderivative of xy squared? Well, the antiderivative of xy squared, I want to make sure I'm color consistent. Well, the antiderivative of x is x to the 1 half, sorry, x squared over 2. And then y squared is just a constant, right? And then we don't have to worry about plus c, since this is a definite integral. And we're going to evaluate that at 2 and 0. And then we still have the outside integral with respect to y. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then y squared is just a constant, right? And then we don't have to worry about plus c, since this is a definite integral. And we're going to evaluate that at 2 and 0. And then we still have the outside integral with respect to y. So once we figure that out, we're going to integrate it from 0 to 1 with respect to dy. Now what does this evaluate? We put a 2 in here. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And then we still have the outside integral with respect to y. So once we figure that out, we're going to integrate it from 0 to 1 with respect to dy. Now what does this evaluate? We put a 2 in here. If you put a 2 in there, you get 2 squared over 2. Well, that's just 4 over 2. So it's 2y squared minus 0 squared over 2 times y squared. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We put a 2 in here. If you put a 2 in there, you get 2 squared over 2. Well, that's just 4 over 2. So it's 2y squared minus 0 squared over 2 times y squared. Well, that's just going to be 0, so it's minus 0. I won't write that down, because hopefully that's a little bit of second nature to you. We just evaluated this at the two endpoints, and I'm short for space. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So it's 2y squared minus 0 squared over 2 times y squared. Well, that's just going to be 0, so it's minus 0. I won't write that down, because hopefully that's a little bit of second nature to you. We just evaluated this at the two endpoints, and I'm short for space. So this evaluated at 2y squared, and now we evaluate the outside integral. 0, 1, dy. And this is an important thing to realize. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We just evaluated this at the two endpoints, and I'm short for space. So this evaluated at 2y squared, and now we evaluate the outside integral. 0, 1, dy. And this is an important thing to realize. When we evaluated this inside integral, remember what we were doing. We were trying to figure out for a given y what the area of this surface was. Well, not this surface, the area under the surface on this kind of for a given y. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And this is an important thing to realize. When we evaluated this inside integral, remember what we were doing. We were trying to figure out for a given y what the area of this surface was. Well, not this surface, the area under the surface on this kind of for a given y. For a given y, that surface kind of turns into a curve. And we've tried to figure out the area under that curve in the traditional sense. So this ended up being a function of y. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Well, not this surface, the area under the surface on this kind of for a given y. For a given y, that surface kind of turns into a curve. And we've tried to figure out the area under that curve in the traditional sense. So this ended up being a function of y. And that makes sense, because depending on which y we pick, we're going to get a different area here. Because obviously, depending on which y we pick, the area, kind of a wall, drops straight down. That area is going to change. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
So this ended up being a function of y. And that makes sense, because depending on which y we pick, we're going to get a different area here. Because obviously, depending on which y we pick, the area, kind of a wall, drops straight down. That area is going to change. So we got a function of y when we evaluated this. And now we just integrate with respect to y. And this is just plain old vanilla definite integration. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
That area is going to change. So we got a function of y when we evaluated this. And now we just integrate with respect to y. And this is just plain old vanilla definite integration. What's the antiderivative of 2y squared? Well, that equals 2 times y to the third over 3, or 2 thirds y to the third. We evaluate that at 1 and 0, which is equal to, let's see, y of 1 to the third times 2 thirds, that's 2 thirds, minus 0 to the third times 2 thirds. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
And this is just plain old vanilla definite integration. What's the antiderivative of 2y squared? Well, that equals 2 times y to the third over 3, or 2 thirds y to the third. We evaluate that at 1 and 0, which is equal to, let's see, y of 1 to the third times 2 thirds, that's 2 thirds, minus 0 to the third times 2 thirds. Well, that's just 0. So it equals 2 thirds. If our units were meters, it would be 2 thirds meters cubed, or cubic meters. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
We evaluate that at 1 and 0, which is equal to, let's see, y of 1 to the third times 2 thirds, that's 2 thirds, minus 0 to the third times 2 thirds. Well, that's just 0. So it equals 2 thirds. If our units were meters, it would be 2 thirds meters cubed, or cubic meters. But there you go. That's how you evaluate a double integral. There really isn't a new skill here. | Double integrals 2 Double and triple integrals Multivariable Calculus Khan Academy.mp3 |
Let's say I have some curve C, and it's described, it can be parametrized, I can't say that word, as let's say x is equal to x of t, y is equal to some function y of t, and let's say that this is valid for t is between a and b. So t is greater than or equal to a, and then less than or equal to b. So if I were to just draw this on a, let me see, I could draw it like this. I'm staying very abstract right now. This is not a very specific example. This is the x-axis, this is the y-axis. My curve, let's say this is when t is equal to a, and then the curve might do something like this. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
I'm staying very abstract right now. This is not a very specific example. This is the x-axis, this is the y-axis. My curve, let's say this is when t is equal to a, and then the curve might do something like this. I don't know what it does. Let's say it's over there. This is t is equal to b. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
My curve, let's say this is when t is equal to a, and then the curve might do something like this. I don't know what it does. Let's say it's over there. This is t is equal to b. This actual point right here will be x of b. That would be the x-coordinate. You evaluate this function at b, and y of b. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
This is t is equal to b. This actual point right here will be x of b. That would be the x-coordinate. You evaluate this function at b, and y of b. And this is, of course, when t is equal to a. The actual coordinate in R2 on the Cartesian coordinates will be x of a, which is this right here, and then y of a, which is that right there. And we've seen that before. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
You evaluate this function at b, and y of b. And this is, of course, when t is equal to a. The actual coordinate in R2 on the Cartesian coordinates will be x of a, which is this right here, and then y of a, which is that right there. And we've seen that before. That's just a standard way of describing a parametric equation or curve using two parametric equations. What I want to do now is describe this same exact curve using a vector-valued function. So if I define a vector-valued function, and if you don't remember what those are, we'll have a little bit of a review here. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
And we've seen that before. That's just a standard way of describing a parametric equation or curve using two parametric equations. What I want to do now is describe this same exact curve using a vector-valued function. So if I define a vector-valued function, and if you don't remember what those are, we'll have a little bit of a review here. Let me say I have a vector-valued function R, and I'll put a little vector arrow on top of it. In a lot of textbooks, they'll just bold it, and they'll leave scalar-valued functions unbolded, but it's hard to draw bold. So I'll put a little vector on top. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So if I define a vector-valued function, and if you don't remember what those are, we'll have a little bit of a review here. Let me say I have a vector-valued function R, and I'll put a little vector arrow on top of it. In a lot of textbooks, they'll just bold it, and they'll leave scalar-valued functions unbolded, but it's hard to draw bold. So I'll put a little vector on top. And let's say that R is a function of t. And these are going to be position vectors. I'm specifying that because, in general, when someone talks about a vector, this vector and this vector are considered equivalent as long as they have the same magnitude and direction. No one really cares about what their start and end points are as long as their direction is the same and their length is the same. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So I'll put a little vector on top. And let's say that R is a function of t. And these are going to be position vectors. I'm specifying that because, in general, when someone talks about a vector, this vector and this vector are considered equivalent as long as they have the same magnitude and direction. No one really cares about what their start and end points are as long as their direction is the same and their length is the same. But when you talk about position vectors, you're saying, no, these vectors are all going to start at 0 at the origin. And when you say it's a position vector, you're implicitly saying this is specifying a unique position. In this case, it's going to be in two-dimensional space, but it could be in three-dimensional space or really even four, five, whatever, n-dimensional space. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
No one really cares about what their start and end points are as long as their direction is the same and their length is the same. But when you talk about position vectors, you're saying, no, these vectors are all going to start at 0 at the origin. And when you say it's a position vector, you're implicitly saying this is specifying a unique position. In this case, it's going to be in two-dimensional space, but it could be in three-dimensional space or really even four, five, whatever, n-dimensional space. So when you say it's a position vector, you're literally saying, OK, this vector literally specifies that point in space. So let's see if we can describe this curve as a position vector-valued function. So we could say r of t, let me switch back to that pink color. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
In this case, it's going to be in two-dimensional space, but it could be in three-dimensional space or really even four, five, whatever, n-dimensional space. So when you say it's a position vector, you're literally saying, OK, this vector literally specifies that point in space. So let's see if we can describe this curve as a position vector-valued function. So we could say r of t, let me switch back to that pink color. I'll just stay in green, is equal to x of t times the unit vector in the x direction. The unit vector gets a little caret on top, a little hat. That's like the arrow for it. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So we could say r of t, let me switch back to that pink color. I'll just stay in green, is equal to x of t times the unit vector in the x direction. The unit vector gets a little caret on top, a little hat. That's like the arrow for it. That just says it's a unit vector. Plus y of t times j. If I was dealing with a curve in three dimensions, I would have plus z of t times k, but we're dealing with two dimensions right here. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
That's like the arrow for it. That just says it's a unit vector. Plus y of t times j. If I was dealing with a curve in three dimensions, I would have plus z of t times k, but we're dealing with two dimensions right here. And so the way this works is you're just taking your, for any t, and still we're going to have t is greater than or equal to a and then less than or equal to b. And this is the exact same thing as that. Let me just redraw it. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
If I was dealing with a curve in three dimensions, I would have plus z of t times k, but we're dealing with two dimensions right here. And so the way this works is you're just taking your, for any t, and still we're going to have t is greater than or equal to a and then less than or equal to b. And this is the exact same thing as that. Let me just redraw it. So let me draw our coordinates right here, our axes. So that's the y-axis and this is the x-axis. So when you evaluate r of a, that's our starting point. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
Let me just redraw it. So let me draw our coordinates right here, our axes. So that's the y-axis and this is the x-axis. So when you evaluate r of a, that's our starting point. So let me do that. So r of a, maybe I'll do it right over here. Our position vector-valued function, evaluated at t is equal to a, is going to be equal to x of a times our unit vector in the x direction plus y of a times our unit vector in the vertical direction, or in the y direction. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So when you evaluate r of a, that's our starting point. So let me do that. So r of a, maybe I'll do it right over here. Our position vector-valued function, evaluated at t is equal to a, is going to be equal to x of a times our unit vector in the x direction plus y of a times our unit vector in the vertical direction, or in the y direction. What's that going to look like? Well, x of a is this thing right here. So it's x of a times the unit vector. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
Our position vector-valued function, evaluated at t is equal to a, is going to be equal to x of a times our unit vector in the x direction plus y of a times our unit vector in the vertical direction, or in the y direction. What's that going to look like? Well, x of a is this thing right here. So it's x of a times the unit vector. So it's really, maybe the unit vector is this long. It has length 1. So now we're just going to have a length of x of a in that direction. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So it's x of a times the unit vector. So it's really, maybe the unit vector is this long. It has length 1. So now we're just going to have a length of x of a in that direction. And then same thing in y of a. It's going to be y of a length in that direction. But the bottom line, this vector right here, if you add these scaled values of these two unit vectors, you're going to get r of a looking something like this. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So now we're just going to have a length of x of a in that direction. And then same thing in y of a. It's going to be y of a length in that direction. But the bottom line, this vector right here, if you add these scaled values of these two unit vectors, you're going to get r of a looking something like this. It's going to be a vector that looks something like that. Just like that. It's a vector. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
But the bottom line, this vector right here, if you add these scaled values of these two unit vectors, you're going to get r of a looking something like this. It's going to be a vector that looks something like that. Just like that. It's a vector. It's a position vector. That's why we're kneeling it at the origin, but drawing it in standard position. And that right there is r of a. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
It's a vector. It's a position vector. That's why we're kneeling it at the origin, but drawing it in standard position. And that right there is r of a. Now, what happens if a increases a little bit? What is r of a plus a little bit? And I don't know, we could call that r of a plus delta, or r of a plus h. We do it in different colors. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
And that right there is r of a. Now, what happens if a increases a little bit? What is r of a plus a little bit? And I don't know, we could call that r of a plus delta, or r of a plus h. We do it in different colors. So let's say we increase a a little bit. r of a plus some small h. Well, that's just going to be x of a plus h times the unit vector i plus y times a plus h times the unit vector j. And what's that going to look like? | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
And I don't know, we could call that r of a plus delta, or r of a plus h. We do it in different colors. So let's say we increase a a little bit. r of a plus some small h. Well, that's just going to be x of a plus h times the unit vector i plus y times a plus h times the unit vector j. And what's that going to look like? Well, we're going to go a little bit further down the curve. That's like saying the coordinate x of a plus h and y plus a plus h. It might be that point right there. So it'll be a new unit vector. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
And what's that going to look like? Well, we're going to go a little bit further down the curve. That's like saying the coordinate x of a plus h and y plus a plus h. It might be that point right there. So it'll be a new unit vector. Sorry, it'll be a new vector, position vector, not a unit vector. These don't necessarily have length 1. That might be right here. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So it'll be a new unit vector. Sorry, it'll be a new vector, position vector, not a unit vector. These don't necessarily have length 1. That might be right here. Let me do that same color as this. So it might be just like that. So that right here is r of a plus h. So you see, as you keep increasing your value of t until you get to b, these position vectors are going to keep specifying points along this curve. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
That might be right here. Let me do that same color as this. So it might be just like that. So that right here is r of a plus h. So you see, as you keep increasing your value of t until you get to b, these position vectors are going to keep specifying points along this curve. So the curve, let me draw the curve in a different color. The curve looks something like this. It's meant to look exactly like the curve that I have up here. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So that right here is r of a plus h. So you see, as you keep increasing your value of t until you get to b, these position vectors are going to keep specifying points along this curve. So the curve, let me draw the curve in a different color. The curve looks something like this. It's meant to look exactly like the curve that I have up here. And for example, r of b is going to be a vector that looks like this. It's going to be a vector that looks like that. Let me draw it relatively straight. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
It's meant to look exactly like the curve that I have up here. And for example, r of b is going to be a vector that looks like this. It's going to be a vector that looks like that. Let me draw it relatively straight. That vector right there is r of b. So hopefully you realize that, look, these position vectors really are specifying the same points on this curve as this original, I guess, straight up parametrization that we did for this curve. And I just want to do that as a little bit of review, because we're now going to break in into the idea of actually taking a derivative of this vector-valued function. | Position vector valued functions Multivariable Calculus Khan Academy.mp3 |
So I think I should probably start off by addressing the elephant in the living room here. I am sadly not Sal, but I'm still gonna teach you some math. My name is Grant. I'm pretty much a math enthusiast. I enjoy making animations of things when applicable, and boy is that applicable when it comes to multivariable calculus. So the first thing we gotta get straight is what is this word multivariable that separates calculus as we know it from the new topic that you're about to study? Well, I could say it's all about multivariable functions. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
I'm pretty much a math enthusiast. I enjoy making animations of things when applicable, and boy is that applicable when it comes to multivariable calculus. So the first thing we gotta get straight is what is this word multivariable that separates calculus as we know it from the new topic that you're about to study? Well, I could say it's all about multivariable functions. That doesn't really answer anything because what's a multivariable function? And basically, the kinds of functions that we're used to dealing with in the old world, in the ordinary calculus world, will have a single input, some kind of number as their input, and then an output's just a single number. And you would call this a single variable function basically because that guy there is the single variable. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
Well, I could say it's all about multivariable functions. That doesn't really answer anything because what's a multivariable function? And basically, the kinds of functions that we're used to dealing with in the old world, in the ordinary calculus world, will have a single input, some kind of number as their input, and then an output's just a single number. And you would call this a single variable function basically because that guy there is the single variable. So then a multivariable function is something that handles multiple variables. So, you know, it's common to write it as like xy. It doesn't really matter what letters to use. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
And you would call this a single variable function basically because that guy there is the single variable. So then a multivariable function is something that handles multiple variables. So, you know, it's common to write it as like xy. It doesn't really matter what letters to use. And it could be, you know, xyz, x1, x2, x3, a whole bunch of things. But just to get started, we often think just two variables. And this will output something that depends on both of those. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
It doesn't really matter what letters to use. And it could be, you know, xyz, x1, x2, x3, a whole bunch of things. But just to get started, we often think just two variables. And this will output something that depends on both of those. Commonly, it'll output just a number. So you might imagine a number that depends on x and y in some way, like x squared plus y. But it could also output a vector, right? | Multivariable functions Multivariable calculus Khan Academy.mp3 |
And this will output something that depends on both of those. Commonly, it'll output just a number. So you might imagine a number that depends on x and y in some way, like x squared plus y. But it could also output a vector, right? So you could also imagine something that's got multivariable input, f of x, y, and it outputs something that also has multiple variables, like, I mean, I'm just making stuff up here, 3x and, you know, 2y. And this isn't set in stone, but the convention is to usually think if there's multiple numbers that go into the output, think of it as a vector. If there's multiple numbers that go into the input, just kind of write them more sideways like this and think of them as a point in space. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
But it could also output a vector, right? So you could also imagine something that's got multivariable input, f of x, y, and it outputs something that also has multiple variables, like, I mean, I'm just making stuff up here, 3x and, you know, 2y. And this isn't set in stone, but the convention is to usually think if there's multiple numbers that go into the output, think of it as a vector. If there's multiple numbers that go into the input, just kind of write them more sideways like this and think of them as a point in space. Because, I mean, when you look at something like this, and you've got an x and you've got a y, you could think about those as two separate numbers. You know, here's your number line with the point x on it somewhere. Maybe that's 5, maybe that's 3, it doesn't really matter. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
If there's multiple numbers that go into the input, just kind of write them more sideways like this and think of them as a point in space. Because, I mean, when you look at something like this, and you've got an x and you've got a y, you could think about those as two separate numbers. You know, here's your number line with the point x on it somewhere. Maybe that's 5, maybe that's 3, it doesn't really matter. And then you've got another number line, and it's y. And you could think of them as separate entities. But it would probably be more accurate to call it multidimensional calculus, because really, instead of thinking of, you know, x and y as separate entities, whenever you see two things like that, you're going to be thinking about the xy-plane and thinking about just a single point. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
Maybe that's 5, maybe that's 3, it doesn't really matter. And then you've got another number line, and it's y. And you could think of them as separate entities. But it would probably be more accurate to call it multidimensional calculus, because really, instead of thinking of, you know, x and y as separate entities, whenever you see two things like that, you're going to be thinking about the xy-plane and thinking about just a single point. And you'd think of this as a function that takes a point to a number, or a point to a vector. And a lot of people, when they start teaching multivariable calculus, they just jump into the calculus. And there's lots of fun things, partial derivatives, gradients, good stuff that you'll learn. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
But it would probably be more accurate to call it multidimensional calculus, because really, instead of thinking of, you know, x and y as separate entities, whenever you see two things like that, you're going to be thinking about the xy-plane and thinking about just a single point. And you'd think of this as a function that takes a point to a number, or a point to a vector. And a lot of people, when they start teaching multivariable calculus, they just jump into the calculus. And there's lots of fun things, partial derivatives, gradients, good stuff that you'll learn. But I think, first of all, I want to spend a couple videos just talking about the different ways we visualize the different types of multivariable functions. So, as a sneak peek, I'm just going to go through a couple of them really quickly right now, just so you kind of whet your appetite and see what I'm getting at. But the next few videos are going to go through them in much, much more detail. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
And there's lots of fun things, partial derivatives, gradients, good stuff that you'll learn. But I think, first of all, I want to spend a couple videos just talking about the different ways we visualize the different types of multivariable functions. So, as a sneak peek, I'm just going to go through a couple of them really quickly right now, just so you kind of whet your appetite and see what I'm getting at. But the next few videos are going to go through them in much, much more detail. So, first of all, graphs. When you have multivariable functions, graphs become three-dimensional. But these only really apply to functions that have some kind of two-dimensional input, which you might think about as living on this xy-plane. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
But the next few videos are going to go through them in much, much more detail. So, first of all, graphs. When you have multivariable functions, graphs become three-dimensional. But these only really apply to functions that have some kind of two-dimensional input, which you might think about as living on this xy-plane. And a single number is their output. And the height of the graph is going to correspond with that output. Like I said, you'll be able to learn much more about that in the dedicated video on it. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
But these only really apply to functions that have some kind of two-dimensional input, which you might think about as living on this xy-plane. And a single number is their output. And the height of the graph is going to correspond with that output. Like I said, you'll be able to learn much more about that in the dedicated video on it. But these functions also can be visualized just in two dimensions, flattening things out, where we visualize the entire input space and associate a color with each point. So this is the kind of thing where you'd have some function that's got a two-dimensional input, it would be f of x, y, and what we're looking at is the xy-plane, all of the input space, and this outputs just some number, maybe it's like x squared, but you know that, and maybe some complicated thing. And the color tells you roughly the size of that output, and the lines here, called contour lines, tell you which inputs all share a constant output value. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
Like I said, you'll be able to learn much more about that in the dedicated video on it. But these functions also can be visualized just in two dimensions, flattening things out, where we visualize the entire input space and associate a color with each point. So this is the kind of thing where you'd have some function that's got a two-dimensional input, it would be f of x, y, and what we're looking at is the xy-plane, all of the input space, and this outputs just some number, maybe it's like x squared, but you know that, and maybe some complicated thing. And the color tells you roughly the size of that output, and the lines here, called contour lines, tell you which inputs all share a constant output value. And again, I'll go into much more detail there. These are really nice, much more convenient than three-dimensional graphs to just sketch out. Moving right along, I'm also going to talk about surfaces in three-dimensional space. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
And the color tells you roughly the size of that output, and the lines here, called contour lines, tell you which inputs all share a constant output value. And again, I'll go into much more detail there. These are really nice, much more convenient than three-dimensional graphs to just sketch out. Moving right along, I'm also going to talk about surfaces in three-dimensional space. They look like graphs, but they actually deal with a much different animal that you could think of it as mapping two dimensions, and I like to sort of spush it about, and we've got kind of a two-dimensional input that somehow moves into three dimensions, and you're just looking at what the output of that looks like, not really caring about how it gets there. These are called parametric surfaces. Another fun one is a vector field, where every input point is associated with some kind of vector, which is the output of the function there, so this would be a function with a two-dimensional input and then two-dimensional output, because each of these are two-dimensional vectors. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
Moving right along, I'm also going to talk about surfaces in three-dimensional space. They look like graphs, but they actually deal with a much different animal that you could think of it as mapping two dimensions, and I like to sort of spush it about, and we've got kind of a two-dimensional input that somehow moves into three dimensions, and you're just looking at what the output of that looks like, not really caring about how it gets there. These are called parametric surfaces. Another fun one is a vector field, where every input point is associated with some kind of vector, which is the output of the function there, so this would be a function with a two-dimensional input and then two-dimensional output, because each of these are two-dimensional vectors. And the fun part with these guys is that you can just kind of imagine a fluid flowing, so here's a bunch of droplets like water, and they kind of flow along that, and that actually turns out to give insight about the underlying function. It's one of those beautiful aspects of multivariable calc, and we'll get lots of exposure to that. Again, I'm just sort of zipping through to whet your appetite. | Multivariable functions Multivariable calculus Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.