Double Integral 1 Introduction to the double integral
Double Integral 1
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- So far, we've used 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.
- Let's call that, you know, this is y is equal to
- 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,
- let's say, x is equal to a and x is equal to b.
- So this is the area I want to figure out.
- 17 00:00:46,87 --> 00:00:50,87 What I do is, I split it up into a bunch of columns
- or a bunch of rectangles.
- Where-- 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 1 of the rectangles.
- Well, the area of the rectangle is just base
- times height, right?
- Well, we're going to make these rectangles really skinny and
- just sum up an infinite number of them.
- So we want 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, at that point.
- It's going to be f of-- if this is x0, 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 just going to be a bunch of them.
- One there, one there.
- Then we'll get the area, and if we have infinite number of
- these rectangles, and 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.
- We're going to take the sums of these rectangles, from x is
- equal to a, to x is equal to b.
- And the sum, or the areas that we're summing up, are going to
- be-- the height is f of x, and the width is d of x.
- 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 to x is equal to b.
- And that's just a little bit of 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 we're thinking in three dimensions, a
- surface is going to be a function of x and y.
- So we can write a surface as, instead of y is a function
- of f and x-- I'm sorry.
- Instead of saying that y is 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.
- So now, before, our domain was just-- at least, you know, for
- most of what we dealt with-- was just the x-axis, or 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 the
- reals right now, I don't want to get too technical.
- And then it'll pop out another number, and if we wanted to
- graph it, it'll be our height.
- And so that could be the 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.
- But as you can see, it's a server.
- It's like a piece of paper that's bent.
- Let's see, 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?
- How do we figure out the volume?
- It seems like a bit of a stretch, given what we've
- learned from this.
- But what if-- and I'm just going to draw an abstract
- surface here-- let me draw some axis.
- Let's see, that's my x-axis.
- 94 00:04:06,39 --> 00:04:09,072 That's my y-axis.
- That's my z-axis.
- I don't practice these videos ahead of time, so I'm often
- wondering what I'm about to draw.
- So that's x, that's y, and that's z.
- And let's say I have some surface.
- I'll just draw something.
- I don't know what it is.
- Some surface.
- This is our surface.
- z is a function of x and y.
- So, for example, you give me a coordinate in the xy plane.
- Say, here, I'll put it into the function and it'll
- give us a z value now.
- And I'll plot it there and it'll 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
- 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.
- And let's say-- so, if we wanted to draw it in the xy
- plane, like you can kind of view the projection of the
- surface of the xy plane, or the shadow of the
- surface of the xy plane.
- What are the bounds?
- You can almost view-- what are the bounds of the domain?
- Well, let's say that this point-- let's say that this
- right here, that's 0, 0 in the xy plane.
- Let's say that this is y is equal to-- I don't know,
- that's 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, it would be a line like that.
- A 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 wanted to figure out the area of--
- let's just say, this sliver.
- Let's say we had a-- well, actually let
- me go the other way.
- Let's say we had a constant y.
- So let's say I had some sliver.
- I don't want to confuse you.
- Let's say that I had some constant y.
- I just want to give you the intuition.
- You know, let's say.
- I don't know what that is.
- It's an arbitrary y.
- But for some constant y, what if I could just figure out the
- area under the curve there?
- How would I figure out just the area under that curve?
- It'll be a function of which y I pick, right?
- Because if I pick a y here, it'll be a different area.
- If I pick a y there, it'll 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's-- 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 is going to be a function of the x
- I have and the y I picked.
- So what would be the area of this sheet of paper?
- It's kind of a constant y.
- It's part of-- it's a sheet of paper within this volume,
- you can kind of view it.
- Well, it would be-- we said the height of each of these
- rectangles is f of xy, right?
- That's the height.
- It depends which x and y we pick down here.
- 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 would look like that. x is going from 0 to b.
- Fair enough.
- And this would actually give us a function of y.
- This would give us an expression so that 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 could I do?
- If I know the area of any given sliver, what if I multiply the
- area of that sliver times dy?
- This is a dy.
- Let me do it in a vibrant color.
- So dy, a very small change in y.
- If I multiplied this area times a small dy, then
- all of a sudden I have a sliver of volume.
- Hopefully that makes some sense.
- I'm making that-- that little cut that I took the area of--
- by making it three 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.
- It gets some depth.
- This little area that I'm shading right here gets
- depth of that sheet.
- Now if I added all of those sheets that now have depth, if
- I took the infinite sum-- so if I took the integral of this
- from my lower y bound-- from 0 to my upper y bound, a, then--
- at least based 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 we're going to do.
- And I think you're going to find out that actually
- calculating the volumes are pretty straightforward,
- 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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