Double Integrals 3 Let's integrate dy first!
Double Integrals 3
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- In the last a video we figured out the volume between this
- surface, which was xy squared and the xy-plane when x went
- from 0 to 2 and y went from 0 to 1.
- And the way we did it is we integrated with
- respect to x first.
- We said, pick a y, and let's just figure out the
- area under the curve.
- And so we integrated with respect to x first, and then we
- integrated with respect to y.
- But we could have done it the other way around.
- So let's do that and just make sure we got the right answer.
- So let me erase a lot of this.
- So remember, our answer was 2/3 when we integrated with respect
- to x first, and then with respect to y.
- But I will show you that we can integrate the other way around.
- That's good when you can get the same answer in
- two different ways.
- So let me redraw that graph because I want to give
- you the intuition again.
- So that's my x-axis, y-axis, z-axis.
- x, y, z.
- Then this is my xy-plane.
- down here.
- y goes from 0 to 1; x goes from 0 to 2, This is x equals 1,
- this is equals 2, this is y equals 1, and then the graph--
- I will do my best to draw it.
- Looks something-- let me get some contrast going here.
- So the graph looks something like this.
- Let me see if I can draw it.
- On this side it looks something like that and then it comes
- down like that, straight.
- And then, the volume we care about.
- It's actually this volume underneath the graph.
- This is the top of the surface on that side.
- We care about this volume underneath the surface.
- And then when we draw the bottom of the surface-- let
- me do it in a darker color-- looks something like this.
- This is the bottom underneath the surface.
- I can even shade it a little bit just to show you that
- it's the underneath.
- Hopefully that's a decent rendering of it.
- Let's look back at what we had before.
- It's like a page that I just flipped up at this point, and
- we care about this volume, kind of the colored
- area under there.
- So let's figure out how to do it.
- Last time we integrated with respect to x first.
- Let's integrate with respect to y first.
- So let's hold x constant.
- So if we hold x constant what we could do is for a given
- x-- let's pick an x.
- So if we pick a given x, let's pick the x here.
- Then what we can do, for a given x, you can view that
- function of x and y.
- If x is a constant, let's say if x is 1 then z is just
- equal to y squared.
- That's easy to figure out the area under because we can see
- that x isn't the constant, but we can treat it as a constant.
- So for example, for any given x, we would have
- a curve like this.
- What we could do is we could try to figure out the
- area of this curve first.
- So how do we do that?
- Well, we just said, we could kind of view this function up
- here as z is equal to xy squared because that's
- exactly what it is.
- But we're holding x constant.
- We're treating it like a constant.
- To figure out that area we could take a dy, a change
- in y, multiply it by the height, which is xy squared.
- So we take xy squared, multiply it by dy, and then if we want
- this entire area we integrate it from y is equal to
- 0 to y is equal to 1.
- Fair enough.
- Now once we have that area, if you want the volume underneath
- this entire surface what we could do is we can multiply
- this area times dx and get some depth going.
- Let me pick a nice color, that's green.
- So that's our dx.
- So if we multiply that times dx we would get some depth.
- Let me do a darker color, get some contrast.
- Sometimes I feel like that guy who paints on PBS.
- So now we have the volume of this, you kind of view it-- the
- area under the curve times a dx, so we have some depth here.
- So it's time dx.
- And if we want to figure out the entire volume under this
- surface-- between the surface and the xy-plane given this
- constraint to our domain-- we just integrate from x
- is equal to 0 to 2.
- All right, so let's think about it.
- This area in green here that we started with, that
- should be a function of x.
- We held x constant, but depending on which x you pick
- this area is going to change.
- So when we evaluate this magenta inner integral with
- respect to y we should get a function of x.
- And then when you evaluate the whole thing we'll
- get our volumes.
- So let's do it.
- Let's evaluate this inner integral.
- Hold x constant.
- What's the antiderivative of y squared?
- It's y of the third over 3.
- The x is a constant, right?
- We're going to evaluate that at 1 and at 0.
- The outer integral is still with respect to x dx.
- This is equal to-- let's see.
- When you evaluate y is equal to 1 you get 1 to the third.
- That's 1.
- So it's x/3 minus when y is 0 then that whole
- thing just becomes 0.
- This purple expression is just x/3.
- And then we still have the outside integral
- from 0 to 2 dx.
- So given what x we have, the area of this green surface--
- that was where we started.
- Given any given x, that area-- I wanted something
- with some contrast.
- This area is x/3 depending on which x you pick.
- If x is 1, this area right here is 1/3.
- But now we're going to integrate underneath the entire
- surface and get our volume.
- And like I said, when you integrate it,
- it's a function of x.
- So let's do that.
- And this is just plain old vanilla, standard integral.
- So what's the antiderivative of x?
- It's x squared over 2.
- We have a 1/3 there so it equals x squared
- over 2 times 3.
- So x squared over 6.
- And we're going to evaluate it at 2 and at 0.
- 2 squared over 6 is 4/6.
- Minus 0/6, which is equal to 0.
- Equals 4/6.
- What is 4/6?
- Well, that's just the same thing as 2/3.
- So the volume under the surface is 2/3, and if you watched
- the previous video you will appreciate the fact that when
- we integrated the other way around, when we did it with
- respect to x first and then y, we got the exact same answer.
- So the universe is in proper working order.
- And I've surprisingly, actually finished this
- video with extra time.
- So for fun, we can just spin this graph and just appreciate
- the fact that we have figured out the volume between this
- surface, xy squared and the xy-plane.
- Pretty neat.
- Anyway, I'll see you in the next video.
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