0:00:00.000,0:00:04.392
PROFESSOR: I have[br]some assignments

0:00:04.392,0:00:06.344
that I want to give you back.

0:00:06.344,0:00:09.510
And I'm just going[br]to put them here,

0:00:09.510,0:00:13.720
and I'll ask you to pick them[br]up as soon as we take a break.

0:00:13.720,0:00:16.850


0:00:16.850,0:00:20.585
There are explanations there[br]how they were computed in red.

0:00:20.585,0:00:23.480
If you have questions,[br]you can as me

0:00:23.480,0:00:26.000
so I can ask my grader about it.

0:00:26.000,0:00:28.950


0:00:28.950,0:00:33.576
Now, I promised you that[br]I would move on today,

0:00:33.576,0:00:35.216
and that's what I'm going to do.

0:00:35.216,0:00:39.480
I'm moving on to something[br]that you're gong to love.

0:00:39.480,0:00:47.844
[? Practically ?] chapter 12[br]is integration of functions

0:00:47.844,0:00:49.320
of several variables.

0:00:49.320,0:00:58.668


0:00:58.668,0:01:01.320
And to warn you[br]we're going to see

0:01:01.320,0:01:08.880
how we introduce introduction[br]to the double integral.

0:01:08.880,0:01:15.521


0:01:15.521,0:01:17.390
But you will say, wait a minute.

0:01:17.390,0:01:22.480
I don't even know if I[br]remember the simple integral.

0:01:22.480,0:01:24.330
And that's why I'm here.

0:01:24.330,0:01:31.520
I want to remind you what the[br]definite integral was both

0:01:31.520,0:01:35.420
as a formal definition-- let's[br]do it as a formal definition

0:01:35.420,0:01:39.930
first, then come up with a[br]geometric interpretation based

0:01:39.930,0:01:40.470
on that.

0:01:40.470,0:01:45.000
And finally write[br]down the definition

0:01:45.000,0:01:49.380
and the fundamental[br]theorem of calculus.

0:01:49.380,0:01:52.390
So assume you have a[br]function that's continuous.

0:01:52.390,0:01:56.310


0:01:56.310,0:02:05.090
Continuous over a certain[br]integral of a, b interval in R.

0:02:05.090,0:02:08.690
And you know that[br]in that case, you

0:02:08.690,0:02:23.040
can "define the[br]definite integral of f

0:02:23.040,0:02:29.476
of x from or between a and b."

0:02:29.476,0:02:35.300
And as the notation is denoted,[br]by integral from a to b f of x

0:02:35.300,0:02:35.800
dx.

0:02:35.800,0:02:42.490


0:02:42.490,0:02:46.095
Well, how do we define this?

0:02:46.095,0:02:47.340
This is just the notation.

0:02:47.340,0:02:51.140
How do we define it?

0:02:51.140,0:02:58.830
We have to have a set up, and[br]we are thinking of a x, y frame.

0:02:58.830,0:03:02.190
You have a function,[br]f, that's continuous.

0:03:02.190,0:03:05.934


0:03:05.934,0:03:07.750
And you are thinking,[br]oh, wait a minute.

0:03:07.750,0:03:11.090
I would like to be[br]able to evaluate

0:03:11.090,0:03:12.466
the area under the integral.

0:03:12.466,0:03:16.590


0:03:16.590,0:03:19.675
And if you ask your teacher[br]when you are in fourth grade,

0:03:19.675,0:03:22.480
your teacher will say, well,[br]I can give you some graphing

0:03:22.480,0:03:23.550
paper.

0:03:23.550,0:03:25.410
And with that[br]graphing paper, you

0:03:25.410,0:03:35.550
can eventually approximate[br]your area like that.

0:03:35.550,0:03:42.790
Sort of what you get here is[br]like you draw a horizontal

0:03:42.790,0:03:46.520
so that the little part[br]above the horizontal

0:03:46.520,0:03:49.100
cancels out with the little[br]part below the horizontal.

0:03:49.100,0:03:51.490
So more or less,[br]the pink rectangle

0:03:51.490,0:03:56.190
is a good approximation[br]of the first slice.

0:03:56.190,0:03:59.660
But you say yeah, but the first[br]slice is a curvilinear slice.

0:03:59.660,0:04:03.080
Yes, but we make it[br]like a stop function.

0:04:03.080,0:04:06.770
So then you say, OK,[br]how about this fellow?

0:04:06.770,0:04:11.230
I'm going to approximate[br]it in a similar way,

0:04:11.230,0:04:15.031
and I'm going to have a bunch[br]of rectangles on this graphing

0:04:15.031,0:04:15.530
paper.

0:04:15.530,0:04:18.450
And I'm going to[br]compute their areas,

0:04:18.450,0:04:20.760
and I'm going to come up[br]with an approximation,

0:04:20.760,0:04:23.872
and I'll give it to my[br]fourth grade teacher.

0:04:23.872,0:04:26.830
And that's what we[br]did in fourth grade,

0:04:26.830,0:04:29.380
but this is not fourth grade.

0:04:29.380,0:04:32.880
And actually, it's[br]very relevant to us

0:04:32.880,0:04:35.610
that this has[br]applications to our life,

0:04:35.610,0:04:38.630
to our digital world,[br]that people did not

0:04:38.630,0:04:44.490
understand when Riemann[br]introduced the Riemann sum.

0:04:44.490,0:04:49.270
They thought, OK, the idea[br]makes sense that practically we

0:04:49.270,0:04:54.220
have a huge picture[br]here, and I'm

0:04:54.220,0:04:59.430
taking a and b and a function[br]that's continuous over a and b.

0:04:59.430,0:05:02.180
And then I say I'm[br]going to split this

0:05:02.180,0:05:08.200
into a equidistant intervals.

0:05:08.200,0:05:10.926
I don't know how[br]many I want, but let

0:05:10.926,0:05:12.089
me make them eight of them.

0:05:12.089,0:05:12.630
I don't know.

0:05:12.630,0:05:14.310
They have to have[br]the same length.

0:05:14.310,0:05:17.410
And I'll call this delta x.

0:05:17.410,0:05:18.510
It has to be the same.

0:05:18.510,0:05:21.782
And, you guys, please forgive[br]me for the horrible picture.

0:05:21.782,0:05:25.990
They don't look like[br]the same step, delta x,

0:05:25.990,0:05:28.630
but it should be the same.

0:05:28.630,0:05:32.645
In each of them I[br]arbitrarily, say it again,

0:05:32.645,0:05:39.340
Magdalena, arbitrarily pick[br]x1 star, and another point,

0:05:39.340,0:05:44.920
x2 star wherever I want inside.

0:05:44.920,0:05:47.580
I'm just getting [INAUDIBLE].

0:05:47.580,0:05:51.300
X4 star, and this is x8 star.

0:05:51.300,0:05:54.130
But let's say that in general[br]I don't know they are 8.

0:05:54.130,0:05:56.270
They could be n.

0:05:56.270,0:05:57.180
xn star.

0:05:57.180,0:05:59.890
And passing to the[br]limit with respect

0:05:59.890,0:06:02.670
to n going to infinity,[br]what am I going to get?

0:06:02.670,0:06:06.800
Well, in the first[br]cam I'm going up,

0:06:06.800,0:06:08.860
and I'm hitting[br]at what altitude?

0:06:08.860,0:06:13.150
I'm hitting at the altitude[br]called f of x1 star.

0:06:13.150,0:06:17.240
And that's going to be the[br]height of this-- what is this?

0:06:17.240,0:06:17.820
Strip?

0:06:17.820,0:06:18.350
Right?

0:06:18.350,0:06:21.365
Or rectangle.

0:06:21.365,0:06:21.865
OK.

0:06:21.865,0:06:24.690
And I'm going to do[br]the same with green

0:06:24.690,0:06:26.990
for the second rectangle.

0:06:26.990,0:06:32.166
I'll pick x2 star, and[br]then that doesn't work.

0:06:32.166,0:06:33.098
And I'll take this.

0:06:33.098,0:06:34.962
Let's see if I can do[br]the light green one,

0:06:34.962,0:06:36.360
because spring is here.

0:06:36.360,0:06:37.292
Let's see.

0:06:37.292,0:06:38.880
That's beautiful.

0:06:38.880,0:06:40.660
I go up.

0:06:40.660,0:06:44.975
I hit here at x2 star.

0:06:44.975,0:06:48.292
I get f of x2 star.

0:06:48.292,0:06:50.310
And so on and so forth.

0:06:50.310,0:06:53.476


0:06:53.476,0:06:57.880
Until I get to, let's say,[br]the last of the Mohicans.

0:06:57.880,0:07:00.610
This will be xn minus[br]1, and this is going

0:07:00.610,0:07:06.190
to be xn star, the purple guy.

0:07:06.190,0:07:07.950
And this is going[br]to be the height

0:07:07.950,0:07:12.280
of that last of the Mohicans.

0:07:12.280,0:07:19.440
So when I compute the sum, I[br]call that approximating sum

0:07:19.440,0:07:23.490
or Riemann approximating sum,[br]because Riemann had nothing

0:07:23.490,0:07:25.880
better to do than invent it.

0:07:25.880,0:07:27.880
He didn't even know[br]that we are going

0:07:27.880,0:07:32.850
to get pixels that are in[br]larger and larger quantities.

0:07:32.850,0:07:36.040
Like, we get 3,000 by 900.

0:07:36.040,0:07:41.440
He didn't know we are going to[br]have all those digital gadgets.

0:07:41.440,0:07:45.645
But passing to the[br]limit practically should

0:07:45.645,0:07:49.330
be easier to understand[br]for teenagers now

0:07:49.330,0:07:53.300
age, because it's like[br]making the number of pixels

0:07:53.300,0:07:57.880
larger and larger, and the[br]pixels practically invisible.

0:07:57.880,0:08:01.710
Remember, I mean, I don't[br]know, those old TVs,

0:08:01.710,0:08:04.240
color TVs where you could[br]still see the squares?

0:08:04.240,0:08:05.154
STUDENT: Mm-hm.

0:08:05.154,0:08:06.070
PROFESSOR: Well, yeah.

0:08:06.070,0:08:07.980
When you were little.

0:08:07.980,0:08:10.850
But I remember them[br]much better than you.

0:08:10.850,0:08:13.970
And, yes, as the number[br]of pixels will increase,

0:08:13.970,0:08:18.190
that means I'm taking the limit[br]and going larger and larger.

0:08:18.190,0:08:20.610
That means[br]practically limitless.

0:08:20.610,0:08:22.720
Infinity will give[br]me an ideal image.

0:08:22.720,0:08:27.125
My eye will be as if I could see[br]the image that's a curvilinear

0:08:27.125,0:08:31.010
image as a real person.

0:08:31.010,0:08:35.039
And, of course, the[br]quality of our movies

0:08:35.039,0:08:36.058
really increased a lot.

0:08:36.058,0:08:41.210
And this is what I'm[br]trying to emphasize here.

0:08:41.210,0:08:47.120
So you have f of x1 star delta[br]x plus the last rectangle

0:08:47.120,0:08:51.580
area, f of xn star delta x.

0:08:51.580,0:08:56.200
Well, as a mathematician,[br]I don't write it like that.

0:08:56.200,0:08:58.770
How do I write it[br]as a mathematician?

0:08:58.770,0:09:00.680
Well, we are funny people.

0:09:00.680,0:09:02.150
We like Greek.

0:09:02.150,0:09:03.050
It's all Greek to me.

0:09:03.050,0:09:16.420
So we go sum and from-- no. k[br]from 1 to n, f of x sub k star.

0:09:16.420,0:09:23.742
So I have k from 1 to n exactly[br]an rectangles area to add.

0:09:23.742,0:09:25.685
And this is going to[br]be [INAUDIBLE], which

0:09:25.685,0:09:28.046
is the same everywhere.

0:09:28.046,0:09:35.730
In that case, I made[br]the partition is equal.

0:09:35.730,0:09:39.190
So practically I have[br]the same distance.

0:09:39.190,0:09:41.100
And what is this[br]limit? [? Lim ?]

0:09:41.100,0:09:45.750
is going to be exactly integral[br]from a to b of f of x dx.

0:09:45.750,0:09:48.825
And I make a smile here,[br]and I say I'm very happy.

0:09:48.825,0:09:55.430
This is as a meaning is[br]the area under the graph.

0:09:55.430,0:09:57.510
If-- well, I didn't[br]say something.

0:09:57.510,0:10:01.390
If I want it to be[br]positive, otherwise it's

0:10:01.390,0:10:04.400
getting not to be the[br]area under the graph.

0:10:04.400,0:10:08.350
The integral will still[br]be defined like that.

0:10:08.350,0:10:12.110
But what's going to happen if[br]I have, for example, half of it

0:10:12.110,0:10:15.380
above and half of it below?

0:10:15.380,0:10:18.180
I'm going to get this,[br]and I'm going to get that.

0:10:18.180,0:10:23.250
And when I add them, I'm going[br]to get a negative answer,

0:10:23.250,0:10:26.500
because this is a negative[br]area, and that's a positive area

0:10:26.500,0:10:28.680
and they try to[br]annihilate each other.

0:10:28.680,0:10:32.010
But this guy under[br]the water is stronger,

0:10:32.010,0:10:35.930
like an iceberg that's[br]20% on tip of the water,

0:10:35.930,0:10:39.080
80% of the iceberg[br]is under the water.

0:10:39.080,0:10:39.830
So the same thing.

0:10:39.830,0:10:45.400
I'm going to get a negative[br]answer in volume [INAUDIBLE].

0:10:45.400,0:10:45.980
OK.

0:10:45.980,0:10:49.080
Now, we remember that[br]very well, but now we

0:10:49.080,0:10:54.650
have to generalize this[br]thingy to something else.

0:10:54.650,0:10:57.470


0:10:57.470,0:11:03.207
And I will give you[br]a curvilinear domain.

0:11:03.207,0:11:04.040
Where shall I erase?

0:11:04.040,0:11:07.290
I don't know.

0:11:07.290,0:11:09.390
Here.

0:11:09.390,0:11:12.500
What if somebody gives you[br]the image of a potatoe-- well,

0:11:12.500,0:11:13.380
I don't know.

0:11:13.380,0:11:14.650
Something.

0:11:14.650,0:11:15.690
A blob.

0:11:15.690,0:11:24.860
Some nice curvilinear domain--[br]and says, you know what?

0:11:24.860,0:11:29.895
I want to approximate the area[br]of this image, curvilinear

0:11:29.895,0:11:35.140
image, to the best[br]of my abilities.

0:11:35.140,0:11:42.370
And compute it, and eventually I[br]have some weighted sum of that.

0:11:42.370,0:11:52.340
So if one would have[br]to compute the area,

0:11:52.340,0:11:55.970
it wouldn't be so hard,[br]because we would say,

0:11:55.970,0:12:05.116
OK, I have to[br]"partition this domain

0:12:05.116,0:12:19.766
into small sections using[br]a rectangular partition

0:12:19.766,0:12:31.450
or square partition."

0:12:31.450,0:12:32.100
And how?

0:12:32.100,0:12:34.805
Well, I'm going to--[br]you have to imagine

0:12:34.805,0:12:41.192
that I have a bunch[br]of a grid, and I'm

0:12:41.192,0:12:43.180
partitioning the whole thing.

0:12:43.180,0:12:53.630


0:12:53.630,0:12:55.730
And you say, wait a minute.

0:12:55.730,0:12:56.530
Wait a minute.

0:12:56.530,0:12:57.690
It's not so easy.

0:12:57.690,0:13:01.500
I mean, they are not all[br]the same area, Magdalena.

0:13:01.500,0:13:05.840
Even if you tried to make these[br]equidistant in both directions,

0:13:05.840,0:13:07.880
look at this guy.

0:13:07.880,0:13:09.070
Look at that guy.

0:13:09.070,0:13:10.810
He's much bigger than that.

0:13:10.810,0:13:14.080
Look at this small[br]guy, and so on.

0:13:14.080,0:13:26.967
So we have to imagine that we[br]look at the so-called normal

0:13:26.967,0:13:27.550
the partition.

0:13:27.550,0:13:34.070


0:13:34.070,0:13:37.480
And let's say in the normal,[br]or the length of the partition,

0:13:37.480,0:13:38.990
is denoted like that.

0:13:38.990,0:13:41.030
We have to give that a meaning.

0:13:41.030,0:13:51.210
Well, let's say "this[br]is the highest diameter

0:13:51.210,0:14:04.146
for all subdomains[br]in the picture."

0:14:04.146,0:14:06.040
And you say, wait a minute.

0:14:06.040,0:14:08.090
But these subdomains[br]should have names.

0:14:08.090,0:14:11.690
Well, they don't have names,[br]but assume they have areas.

0:14:11.690,0:14:16.385
This would be-- I have to[br]find a way to denote them

0:14:16.385,0:14:18.430
and be orderly.

0:14:18.430,0:14:32.340
A1, A2, A3, A4, A5, AN,[br]AM, AN, stuff like that.

0:14:32.340,0:14:38.440
So practically I'm looking[br]at the highest diameter.

0:14:38.440,0:14:43.290
When I have a domain, I[br]look at the largest instance

0:14:43.290,0:14:44.900
inside that domain.

0:14:44.900,0:14:47.050
So what would be the diameter?

0:14:47.050,0:14:50.110
The largest distance between[br]two points in that domain.

0:14:50.110,0:14:52.265
I'll call that the diameter.

0:14:52.265,0:14:52.765
OK.

0:14:52.765,0:14:57.550
I want that diameter to[br]go got 0 in the limit.

0:14:57.550,0:15:03.360
So I want this partition[br]to go to 0 in the limit.

0:15:03.360,0:15:05.748
And that means I'm[br]"shrinking" the pixels.

0:15:05.748,0:15:08.712


0:15:08.712,0:15:11.182
"Shrinking" in[br]quotes, the pixels.

0:15:11.182,0:15:17.120


0:15:17.120,0:15:20.710
How would I mimic[br]what I did here?

0:15:20.710,0:15:23.435
Well, it would be[br]easier to get the area.

0:15:23.435,0:15:29.410
In this case, I would have[br]some sort of A sum limit.

0:15:29.410,0:15:30.100
I'm sorry.

0:15:30.100,0:15:36.340
The curvilinear[br]area of the domain.

0:15:36.340,0:15:40.570
Let's call it-- what[br]do you want to call it?

0:15:40.570,0:15:45.300
D for domain--[br]inside the domain.

0:15:45.300,0:15:46.208
OK?

0:15:46.208,0:15:48.476
This whole thing would be what?

0:15:48.476,0:16:01.240
Would be limit of summation of,[br]let's say, limit of what kind?

0:16:01.240,0:16:04.000
k from 1 to n.

0:16:04.000,0:16:06.270
Limit n goes to infinity.

0:16:06.270,0:16:16.270
K from 1 to n of[br]these tiny A sub k's,

0:16:16.270,0:16:17.400
areas of the subdomain.

0:16:17.400,0:16:24.520


0:16:24.520,0:16:25.293
Wait a minute.

0:16:25.293,0:16:29.970
But you say, but what if[br]I want something else?

0:16:29.970,0:16:34.080
Like, I'm going to[br]build some geography.

0:16:34.080,0:16:35.110
This is the domain.

0:16:35.110,0:16:38.560
That's something like[br]on a map, and I'm

0:16:38.560,0:16:40.670
going to build a[br]mountain on top of it.

0:16:40.670,0:16:43.400
I'll take some Play-Do,[br]I'll take some Play-Do,

0:16:43.400,0:16:46.266
and I'm going to[br]model some geography.

0:16:46.266,0:16:47.390
And you say, wait a minute.

0:16:47.390,0:16:49.390
Do you make mountains?

0:16:49.390,0:16:52.150
I'm afraid to make Rocky[br]Mountains, because they

0:16:52.150,0:16:55.850
may have points where the[br]function is not smooth.

0:16:55.850,0:16:58.310
If I don't have[br]derivative at the peak,

0:16:58.310,0:17:01.260
them I'm in trouble, in general.

0:17:01.260,0:17:03.020
Although you say,[br]well, but the function

0:17:03.020,0:17:04.740
has to be only continuous.

0:17:04.740,0:17:05.260
I know.

0:17:05.260,0:17:05.760
I know.

0:17:05.760,0:17:09.980
But I don't want any kind[br]of really nasty singularity

0:17:09.980,0:17:12.300
where I can have a[br]crack in the mountain

0:17:12.300,0:17:15.829
or a well or[br]something like that.

0:17:15.829,0:17:18.589
So I assume the[br]geography to be smooth,

0:17:18.589,0:17:21.420
the function of[br][INAUDIBLE] is continuous,

0:17:21.420,0:17:23.310
and the picture[br]should look something

0:17:23.310,0:17:27.654
like-- let's see[br]if I can do that.

0:17:27.654,0:17:33.960


0:17:33.960,0:17:38.170
The projection, the[br]shadow of this geography,

0:17:38.170,0:17:43.395
would be the domain, [? D. ?][br]And this is equal, f of x what?

0:17:43.395,0:17:44.550
You say, what?

0:17:44.550,0:17:46.380
Magdalena, I don't understand.

0:17:46.380,0:17:51.720
The exact shadow of this fellow[br]where I have the sun on top

0:17:51.720,0:17:54.360
here-- that's the sun.

0:17:54.360,0:17:59.330
Spring is coming-- the shade[br]is the plain, or domain, x, y.

0:17:59.330,0:18:03.280
I take all my points in x, y.

0:18:03.280,0:18:05.630
I mean, I take really[br]all my points in x, y,

0:18:05.630,0:18:10.355
and the value of the altitude[br]on this geography at the point

0:18:10.355,0:18:13.676
x, y would be z[br]equals f of x, y.

0:18:13.676,0:18:20.775
And somebody's asking me, OK,[br]if this would be a can of Coke,

0:18:20.775,0:18:23.900
it would be easy to[br]compute the volume, right?

0:18:23.900,0:18:27.680
Practically you have a[br]constant altitude everywhere,

0:18:27.680,0:18:30.150
and you have the area of[br]the base times the height,

0:18:30.150,0:18:32.750
and that's your volume.

0:18:32.750,0:18:39.460
But what if somebody asks you to[br]find the volume under the hat?

0:18:39.460,0:18:47.115
"Find the volume[br]undo this graph."

0:18:47.115,0:18:51.027
STUDENT: I would take it[br]more as two functions.

0:18:51.027,0:18:53.472
So the top line would[br]be the one function,

0:18:53.472,0:18:55.428
and the bottom line would[br]be another function.

0:18:55.428,0:18:58.777
So if you take the volume of the[br]top function minus the volume

0:18:58.777,0:19:00.318
of the bottom[br]function, it'd give you

0:19:00.318,0:19:02.780
the total volume of the object.

0:19:02.780,0:19:05.570
PROFESSOR: And actually,[br]I want the total volume

0:19:05.570,0:19:07.660
above the sea level.

0:19:07.660,0:19:12.750
So I'm going to--[br]sometimes I can take it up

0:19:12.750,0:19:16.115
to a certain level where-- let's[br]say the mountain is up to here,

0:19:16.115,0:19:18.490
and I want it only up to here.

0:19:18.490,0:19:22.220
So I want everything,[br]including the-- the walls

0:19:22.220,0:19:24.137
would be cylindrical.

0:19:24.137,0:19:24.720
STUDENT: Yeah.

0:19:24.720,0:19:26.140
PROFESSOR: If I[br]want all the volume,

0:19:26.140,0:19:27.764
that's going to be[br]a little bit easier.

0:19:27.764,0:19:29.470
Let's see why.

0:19:29.470,0:19:31.703
I will have limit.

0:19:31.703,0:19:35.631
The idea is, as you[br]said very well, limit.

0:19:35.631,0:19:37.104
n goes to infinity.

0:19:37.104,0:19:42.150
A sum k from 1 to n.

0:19:42.150,0:19:44.880
And what kind of[br]partition can I build?

0:19:44.880,0:19:47.640
I'll take the[br]line, and I'll say,

0:19:47.640,0:19:53.030
I'll build myself[br]a partition with a,

0:19:53.030,0:19:57.510
let's say, the[br]typical domain, AK.

0:19:57.510,0:20:00.960
I have A1, A2 A3, A4, AK, AN.

0:20:00.960,0:20:03.473
How may of those little domains?

0:20:03.473,0:20:04.455
AN.

0:20:04.455,0:20:07.940
That will be all the[br]little subdomains

0:20:07.940,0:20:12.410
inside the green curve.

0:20:12.410,0:20:14.510
The green loop.

0:20:14.510,0:20:16.660
In that case, what do I do?

0:20:16.660,0:20:24.260
For each of these guys, I go[br]up, and I go, oh, my god, this

0:20:24.260,0:20:27.380
looks like a skyscraper,[br]but the corners,

0:20:27.380,0:20:29.085
when I go through[br]this surface, are

0:20:29.085,0:20:30.900
in the different dimensions.

0:20:30.900,0:20:32.430
What am I going to do?

0:20:32.430,0:20:34.760
That forces me to[br]build a skyscraper

0:20:34.760,0:20:38.570
by thinking I take a[br]point in the domain,

0:20:38.570,0:20:45.110
I go up until that hits the[br]surface, pinches the surface,

0:20:45.110,0:20:47.620
and this is the[br]altitude that I'm going

0:20:47.620,0:20:50.280
to select for my skyscraper.

0:20:50.280,0:20:54.490
And here I'm going to have[br]another skyscraper, and here

0:20:54.490,0:20:57.800
another one and another one,[br]so practically it's dense.

0:20:57.800,0:21:02.450
I have a skyscraper next[br]to the other or a less like

0:21:02.450,0:21:02.960
[INAUDIBLE].

0:21:02.960,0:21:06.550
Not so many gaps[br]in certain areas.

0:21:06.550,0:21:12.640
So I'm going to say[br]f of x kappa star.

0:21:12.640,0:21:18.680
Now those would be the[br]altitudes of the buildings.

0:21:18.680,0:21:20.920
Magdalena, you don't[br]know how to spell.

0:21:20.920,0:21:27.972
Altitudes of the buildings.

0:21:27.972,0:21:31.860


0:21:31.860,0:21:32.695
What are they?

0:21:32.695,0:21:34.055
Parallel [INAUDIBLE] by P's.

0:21:34.055,0:21:36.230
Can you say parallel by P?

0:21:36.230,0:21:37.040
OK.

0:21:37.040,0:21:40.722
[INAUDIBLE] what.

0:21:40.722,0:21:49.720
Ak where Ak will be the basis[br]of the area of the basis.

0:21:49.720,0:21:51.670
is of my building.

0:21:51.670,0:21:54.860


0:21:54.860,0:21:55.490
OK.

0:21:55.490,0:21:59.290
The green part will[br]be the flat area

0:21:59.290,0:22:03.580
of the floor of the skyscraper.

0:22:03.580,0:22:05.600
Is this hard?

0:22:05.600,0:22:06.830
Gosh, yes.

0:22:06.830,0:22:12.713
If you want to do it by[br]hand and take the limit

0:22:12.713,0:22:15.620
you would really kill[br]yourself in the process.

0:22:15.620,0:22:17.420
This is how you introduce it.

0:22:17.420,0:22:21.900
You can prove this limit exists,[br]and you can prove that limits

0:22:21.900,0:22:32.805
exist and will be the volume of[br]the region under the geography

0:22:32.805,0:22:38.760
z equals f of x,y and[br]above the sea level.

0:22:38.760,0:22:43.233


0:22:43.233,0:22:46.779
The seal level[br]meaning z equals z.

0:22:46.779,0:22:49.717
STUDENT: What's under a of k?

0:22:49.717,0:22:50.300
PROFESSOR: Ak.

0:22:50.300,0:22:51.466
STUDENT: What is [INAUDIBLE]

0:22:51.466,0:22:54.078
PROFESSOR: Volume of the region.

0:22:54.078,0:22:55.745
STUDENT: Oh, I know,[br]like what under it?

0:22:55.745,0:22:56.411
PROFESSOR: Here?

0:22:56.411,0:22:57.259
STUDENT: No, up.

0:22:57.259,0:22:58.047
PROFESSOR: Here?

0:22:58.047,0:22:58.588
STUDENT: Yes.

0:22:58.588,0:23:00.605
PROFESSOR: Area of the[br]basis of a building.

0:23:00.605,0:23:01.563
STUDENT: Oh, the basis.

0:23:01.563,0:23:04.070
PROFESSOR: So practically[br]this green thingy

0:23:04.070,0:23:10.730
is a basis like the base rate.

0:23:10.730,0:23:13.400
How large is the basement[br]of that building.

0:23:13.400,0:23:16.240
Ak.

0:23:16.240,0:23:18.470
Now how am I going[br]to write this?

0:23:18.470,0:23:19.400
This is something new.

0:23:19.400,0:23:27.280
We have to invent a notion[br]for it, and since it's Ak,

0:23:27.280,0:23:31.390
looks more or less like[br]a square or a rectangle.

0:23:31.390,0:23:35.120
You think, well, wouldn't--[br]OK, if it's a rectangle,

0:23:35.120,0:23:38.410
I know I'm going to get[br]delta x and delta y right?

0:23:38.410,0:23:41.535
The width times the height,[br]whatever those two dimensions.

0:23:41.535,0:23:42.400
It makes sense.

0:23:42.400,0:23:44.730
But what if I have[br]this domain that's

0:23:44.730,0:23:47.270
curvilinear or that[br]domain or that domain.

0:23:47.270,0:23:49.690
Of course, the diameter[br]of such a domain

0:23:49.690,0:23:54.000
is less than the diameter of the[br]partition, so I'm very happy.

0:23:54.000,0:23:55.870
The highest diameter,[br]say I can get it here,

0:23:55.870,0:23:59.416
and this is shrinking[br]to zero, and pixels

0:23:59.416,0:24:01.580
are shrinking to zero.

0:24:01.580,0:24:05.410
But what am I going to[br]do about those guys?

0:24:05.410,0:24:09.720
Well, you can assume that[br]I am still approximating

0:24:09.720,0:24:14.750
with some squares and as[br]the pixels are getting

0:24:14.750,0:24:17.315
to be many, many,[br]many more, it doesn't

0:24:17.315,0:24:19.520
matter that I'm doing this.

0:24:19.520,0:24:21.797
Let me show you what I'm doing.

0:24:21.797,0:24:29.070
So on the floor, on the-- this[br]is the city floor, whatever.

0:24:29.070,0:24:32.180
What we do in practice,[br]we approximate that

0:24:32.180,0:24:42.270
like on the graphing paper[br]with tiny square domains,

0:24:42.270,0:24:48.800
and we call them delta Ak will[br]be delta Sk times delta Yk,

0:24:48.800,0:24:53.640
and I tried to make it a uniform[br]partition as much as I can.

0:24:53.640,0:24:56.230
Now as the number of[br]pixels goes to infinity

0:24:56.230,0:24:59.360
and those pixels will[br]become smaller and smaller,

0:24:59.360,0:25:04.240
it doesn't there that the actual[br]contour of your Riemann sum

0:25:04.240,0:25:07.090
will look like graphing paper.

0:25:07.090,0:25:10.410
It will get refined, more[br]refined, more refined, smoother

0:25:10.410,0:25:12.710
and smoother, and[br]it's going to be

0:25:12.710,0:25:17.890
really close to the ideal[br]image, which is a curve.

0:25:17.890,0:25:20.342
So as that end goes[br]to infinity, you're

0:25:20.342,0:25:24.940
not going to see this-- what is[br]this called-- zig zag thingy.

0:25:24.940,0:25:25.810
Not anymore.

0:25:25.810,0:25:31.950
The zig zag thingy will go into[br]the limit to the green curve.

0:25:31.950,0:25:34.566
This is what the[br]pixels are about.

0:25:34.566,0:25:38.300
This is how our[br]life changed a lot.

0:25:38.300,0:25:39.070
OK?

0:25:39.070,0:25:39.840
All right.

0:25:39.840,0:25:41.850
Now good.

0:25:41.850,0:25:45.020
How am I going[br]compute this thing?

0:25:45.020,0:25:47.730


0:25:47.730,0:25:52.430
Well, I don't know, but let[br]me give it a name first.

0:25:52.430,0:25:56.130
It's going to be double[br]integral over-- what

0:25:56.130,0:25:58.960
do want the floor to be called?

0:25:58.960,0:26:02.271


0:26:02.271,0:26:04.440
We called d domain before.

0:26:04.440,0:26:06.460
What should I call this?

0:26:06.460,0:26:09.870
Big D. Not round.

0:26:09.870,0:26:13.870
Over D. That's the[br]floor, the foundation

0:26:13.870,0:26:16.655
of the whole city-- of[br]the whole area of the city

0:26:16.655,0:26:18.050
that I'm looking at.

0:26:18.050,0:26:28.190
Then I have f of xy,[br]da, and what is this?

0:26:28.190,0:26:29.640
This is exactly that.

0:26:29.640,0:26:35.370
It's the limit of sum of the--[br]what is the difference here?

0:26:35.370,0:26:37.040
You say, wait a[br]minute, Magdalena,

0:26:37.040,0:26:40.160
but I think I don't[br]understand what you did.

0:26:40.160,0:26:43.930
You tried to copy the[br]concept from here,

0:26:43.930,0:26:47.630
but you forgot you have a[br]function of two variables.

0:26:47.630,0:26:52.560
In that case, this mister,[br]whoever it is that goes up

0:26:52.560,0:26:57.820
is not xk, it's XkYk.

0:26:57.820,0:27:02.060
So I have two variables--[br]doesn't change anything

0:27:02.060,0:27:03.880
for the couple.

0:27:03.880,0:27:08.190
This couple represents a[br]point on the skyscraper

0:27:08.190,0:27:16.320
so that when I go up, I hit the[br]roof with this exact altitude.

0:27:16.320,0:27:19.670
So what is the double integral[br]of a continuous function

0:27:19.670,0:27:26.480
f of x and y, two variables,[br]with respect to area level.

0:27:26.480,0:27:32.780
Well, it's going to be just[br]the limit of this huge thing.

0:27:32.780,0:27:37.670
In fact, it's how[br]do we compute it?

0:27:37.670,0:27:40.580
Let's see how we[br]compute it in practice.

0:27:40.580,0:27:42.821
It shouldn't be a big deal.

0:27:42.821,0:27:56.078


0:27:56.078,0:27:57.970
What if I have a[br]rectangular domain,

0:27:57.970,0:28:00.855
and that's going to[br]make my life easier.

0:28:00.855,0:28:05.840
I'm going to have a[br]rectangular domain in plane,

0:28:05.840,0:28:07.880
and which one is the x-axis?

0:28:07.880,0:28:09.550
This one.

0:28:09.550,0:28:14.750
From A to B, I have the x[br]moving between a and Mr. y

0:28:14.750,0:28:19.550
says, I'm going to[br]be between c and d.

0:28:19.550,0:28:23.180
C is here, and d is here.

0:28:23.180,0:28:28.480
So this is going to be[br]the so-called rectangle

0:28:28.480,0:28:37.150
a, b cross c, d meaning[br]the set of all the pairs--

0:28:37.150,0:28:42.020
or the couples xy-- inside[br]it, what does it mean?

0:28:42.020,0:28:45.560
x, y you playing with[br]the property there.

0:28:45.560,0:28:48.510
X is between a and b, thank god.

0:28:48.510,0:28:50.210
It's easy.

0:28:50.210,0:28:53.530
And y must be between[br]c and d, also easy.

0:28:53.530,0:28:57.885
A, b, c, d are fixed real[br]numbers in this order.

0:28:57.885,0:29:02.054
A is less than b, and c is less.

0:29:02.054,0:29:05.030
And we have this[br]geography on top,

0:29:05.030,0:29:09.080
and I will tell you[br]what it looks like.

0:29:09.080,0:29:13.720
I'm going to try and draw[br]some beautiful geography.

0:29:13.720,0:29:19.620
And now I'm thinking[br]of my son, who is 10.

0:29:19.620,0:29:23.590
He played with this kind of toy[br]that was exactly this color,

0:29:23.590,0:29:25.795
lime, and it had needles.

0:29:25.795,0:29:27.965
Do you guys remember that toy?

0:29:27.965,0:29:30.910
I am sure you're young[br]enough to remember that.

0:29:30.910,0:29:35.330
You have your palm like that,[br]and you see this square thingy,

0:29:35.330,0:29:37.430
and it's all made[br]of needles that

0:29:37.430,0:29:40.810
look like thin,[br]tiny skyscrapers,

0:29:40.810,0:29:47.400
and you push through and all[br]those needles go up and take

0:29:47.400,0:29:49.660
the shape of your hand.

0:29:49.660,0:29:51.925
And of course, he would[br]put it on his face,

0:29:51.925,0:29:54.350
and you could see[br]his face and so on.

0:29:54.350,0:29:56.100
But what is that?

0:29:56.100,0:29:59.890
That's exactly the Riemann[br]sum, the Riemann approximation,

0:29:59.890,0:30:02.200
because if you think of[br]all those needles or tiny--

0:30:02.200,0:30:07.300
what are they, like[br]the tiny skyscrapers--

0:30:07.300,0:30:11.610
the sum of the them approximates[br]the curvilinear shape.

0:30:11.610,0:30:15.890
If you put that over your face,[br]your face is nice and smooth,

0:30:15.890,0:30:19.190
curvilinear except for[br]a few single areas,

0:30:19.190,0:30:24.600
but if you actually[br]look at that needle

0:30:24.600,0:30:27.640
thingy that is[br]giving the figure,

0:30:27.640,0:30:30.060
you recognize the figure.

0:30:30.060,0:30:33.560
It's like a pattern recognition,[br]but it's not your face.

0:30:33.560,0:30:34.910
I mean it is and it's not.

0:30:34.910,0:30:39.220
It's an approximation of[br]your face, a very rough face.

0:30:39.220,0:30:42.140
You have to take that[br]rough model of your face

0:30:42.140,0:30:43.850
and smooth it out.

0:30:43.850,0:30:44.450
How?

0:30:44.450,0:30:49.930
By passing to the[br]limit, and this is what

0:30:49.930,0:30:52.240
animation is doing actually.

0:30:52.240,0:30:55.915
On top of that you want this[br]to have some other properties--

0:30:55.915,0:31:01.250
illumination of some sort--[br]light coming from what angle.

0:31:01.250,0:31:04.910
That is all rendering[br]techniques are actually

0:31:04.910,0:31:06.810
applied mathematics.

0:31:06.810,0:31:09.550
In animation, the[br]people who programmed

0:31:09.550,0:31:12.940
Toy Story-- that[br]was a long time ago,

0:31:12.940,0:31:16.760
but everything that[br]came after Toy Story 2

0:31:16.760,0:31:20.550
was based on mathematical[br]rendering techniques.

0:31:20.550,0:31:23.825
Everything based on[br]the notion of length.

0:31:23.825,0:31:25.078
All right.

0:31:25.078,0:31:28.012
So the way we compute[br]this in practice

0:31:28.012,0:31:31.081
is going to be very simple,[br]because you're going to think,

0:31:31.081,0:31:33.330
how am I going to do the[br]rectangle for the rectangles?

0:31:33.330,0:31:35.800
That'll be very easy.

0:31:35.800,0:31:43.570
I split the rectangle perfectly[br]into other tiny rectangle.

0:31:43.570,0:31:46.890
Every rectangle will[br]have the same dimension.

0:31:46.890,0:31:48.670
Delta x and delta y.

0:31:48.670,0:31:51.515


0:31:51.515,0:31:53.040
Does it makes sense?

0:31:53.040,0:31:55.920
So practically when[br]I go to the limit,

0:31:55.920,0:32:02.220
I have summation f[br]of xk star, yk star

0:32:02.220,0:32:06.840
inside the delta x[br]delta y delta Magdalena,

0:32:06.840,0:32:11.530
the same kind of displacement[br]when I take k from 1 to n,

0:32:11.530,0:32:16.170
and I pass to the limit[br]according to the partition,

0:32:16.170,0:32:17.840
what's going to happen?

0:32:17.840,0:32:21.240
These guys, according[br]to Mr. Linux,

0:32:21.240,0:32:26.120
will go to be infinitesimal[br]elements, dx, dy.

0:32:26.120,0:32:29.530
This whole thing will[br]go to double integral

0:32:29.530,0:32:36.660
of f of x and y,[br]and Mr. y says, OK

0:32:36.660,0:32:39.330
it's like you want him to[br]integrate him one at a time.

0:32:39.330,0:32:43.612
This is actually something that[br]we are going to see in a second

0:32:43.612,0:32:45.030
and verify it.

0:32:45.030,0:32:50.420
X goes between a and b,[br]and y goes between c and d,

0:32:50.420,0:32:55.008
and this is an application[br]of a big theorem called

0:32:55.008,0:33:01.450
Fubini's Theorem that[br]says, wait a minute,

0:33:01.450,0:33:06.830
if you do it like this over[br]a rectangle a,b cross c,d,

0:33:06.830,0:33:12.360
you're double integral can[br]be written as three things.

0:33:12.360,0:33:18.210
Double integral over your[br]square domain f of x,y dA,

0:33:18.210,0:33:22.140
or you integral from c to[br]d, integral from a to b,

0:33:22.140,0:33:28.170
f of x,y dx dy, or you[br]can also swap the order,

0:33:28.170,0:33:32.142
because you say, well, you can[br]do the integration with respect

0:33:32.142,0:33:34.770
to y first.

0:33:34.770,0:33:36.800
Nobody stops you[br]from doing that,

0:33:36.800,0:33:40.832
and y has to be[br]between what and what?

0:33:40.832,0:33:41.540
STUDENT: C and d.

0:33:41.540,0:33:42.915
PROFESSOR: C and d, thank you.

0:33:42.915,0:33:47.190
And then whatever you get,[br]you get to integrate that

0:33:47.190,0:33:52.170
with respect to x from a to b.

0:33:52.170,0:33:56.280
So no matter in what[br]order you do it,

0:33:56.280,0:33:59.200
you'll get the same thing.

0:33:59.200,0:34:02.990
Let's see an easy example,[br]and you'll say, well,

0:34:02.990,0:34:06.320
start with some [INAUDIBLE][br]example, Magdalena,

0:34:06.320,0:34:08.610
because we are just[br]starting, and that's

0:34:08.610,0:34:10.090
exactly what I'm going to.

0:34:10.090,0:34:11.639
I will just misbehave.

0:34:11.639,0:34:14.630
I'm not going to go by the book.

0:34:14.630,0:34:19.409
And I will say I'm going[br]by whatever I want to go.

0:34:19.409,0:34:27.590
X is between 0, 2, and[br]y is between 0 and 2

0:34:27.590,0:34:32.969
and 3-- this is 2, this[br]is 3-- and my domain

0:34:32.969,0:34:38.630
will be the rectangle[br]0, 2 times 0, 3.

0:34:38.630,0:34:42.161
This is neat on the floor.

0:34:42.161,0:34:59.940
Compute the volume of the[br]box of basis d and height 5.

0:34:59.940,0:35:02.070
Can I draw that?

0:35:02.070,0:35:03.610
It gets out of the picture.

0:35:03.610,0:35:04.570
I'm just kidding.

0:35:04.570,0:35:07.830
This is 5, and that's[br]sort of the box.

0:35:07.830,0:35:10.690


0:35:10.690,0:35:13.590
And you say, wait a minute, I[br]know that from third grade--

0:35:13.590,0:35:16.040
I mean, first grade, whenever.

0:35:16.040,0:35:17.230
How do we do that?

0:35:17.230,0:35:21.340
We go 2 units times[br]3 units that's

0:35:21.340,0:35:25.460
going to be 6 square inches[br]on the bottom of the box,

0:35:25.460,0:35:27.586
and then times 5.

0:35:27.586,0:35:31.437
So the volume has to be[br]2 times 3 times 5, which

0:35:31.437,0:35:35.490
is 30 square inches.

0:35:35.490,0:35:37.090
I don't care what it is.

0:35:37.090,0:35:39.370
I'm a mathematician, right?

0:35:39.370,0:35:39.870
OK.

0:35:39.870,0:35:44.000
How does somebody who just[br]learned Tonelli's-- Fubini

0:35:44.000,0:35:46.964
Tonelli's Theorem[br]do the problem.

0:35:46.964,0:35:49.180
That person will[br]say, wait a minute,

0:35:49.180,0:35:54.800
now I know that the[br]function is going to be z

0:35:54.800,0:36:00.430
equals f of xy, which in[br]this case happens to be cost.

0:36:00.430,0:36:05.020
According to what you told us,[br]the theorem you claim Magdalena

0:36:05.020,0:36:07.130
proved to this theorem,[br]but there is a sketch

0:36:07.130,0:36:08.954
of the proof in the book.

0:36:08.954,0:36:13.360
According to this,[br]the double integral

0:36:13.360,0:36:21.400
that you have over the[br]domain d, and this is dA.

0:36:21.400,0:36:30.530
DA will be called element of[br]area, which is also dx dy.

0:36:30.530,0:36:34.510
This can be solved in[br]two different ways.

0:36:34.510,0:36:38.285
You take integral[br]from-- where is x going?

0:36:38.285,0:36:41.970
Do we want to do it[br]first in x or in y?

0:36:41.970,0:36:44.925
If we put dy dx, that means[br]we integrate with respect

0:36:44.925,0:36:49.250
to y first, and y[br]goes between 0 and 3,

0:36:49.250,0:36:53.200
so I have to pay attention[br]to the limits of integration.

0:36:53.200,0:36:56.290
And then x between[br]0 and 2 and again

0:36:56.290,0:36:58.850
I have to pay attention to[br]the limits of integration

0:36:58.850,0:37:03.121
all the time and,[br]here, who is my f?

0:37:03.121,0:37:06.670
Is the altitude 5 that's[br]constant in my case?

0:37:06.670,0:37:08.500
I'm not worried about it.

0:37:08.500,0:37:10.640
Let me see if I get 30?

0:37:10.640,0:37:16.490
I'm just checking if this[br]theorem was true or is just

0:37:16.490,0:37:20.981
something that you cannot apply.

0:37:20.981,0:37:25.618
How do you integrate[br]5 with respect to y?

0:37:25.618,0:37:26.479
STUDENT: 5y.

0:37:26.479,0:37:27.520
PROFESSOR: 5y, very good.

0:37:27.520,0:37:34.120
So it's going to be 5y between[br]y equals 0 down and y equals 3

0:37:34.120,0:37:38.745
up, and how much[br]is that 5y, we're

0:37:38.745,0:37:41.440
doing y equals 0 down[br]and y equals 3 up,

0:37:41.440,0:37:42.910
what number is that?

0:37:42.910,0:37:44.870
STUDENT: 25.

0:37:44.870,0:37:45.850
PROFESSOR: What?

0:37:45.850,0:37:46.830
STUDENT: 25.

0:37:46.830,0:37:47.810
PROFESSOR: 25?

0:37:47.810,0:37:49.770
STUDENT: One [INAUDIBLE] 15.

0:37:49.770,0:37:54.200
PROFESSOR: No, you did--[br]you are thinking ahead.

0:37:54.200,0:38:00.010
So I go 5 times 3 minus[br]5 times 0 equals 15.

0:38:00.010,0:38:04.110
So when I compute this[br]variation of 5y between y

0:38:04.110,0:38:06.816
equals 3 and y equals[br]0, I just block in

0:38:06.816,0:38:08.260
and make the difference.

0:38:08.260,0:38:09.730
Why do I do that?

0:38:09.730,0:38:15.790
It's the simplest application[br]of that FT, fundamental theorem.

0:38:15.790,0:38:19.480
The one that I did not[br]specify in [INAUDIBLE].

0:38:19.480,0:38:23.490
I should have specified when[br]I have a g function that

0:38:23.490,0:38:28.190
is continuous between[br]alpha and beta, how do we

0:38:28.190,0:38:30.300
integrate with respect to x?

0:38:30.300,0:38:33.485
I get the antiderivative[br]of rule G. Let's call

0:38:33.485,0:38:37.140
that big G. Compute[br]it at the end points,

0:38:37.140,0:38:39.190
and I make the difference.

0:38:39.190,0:38:41.789
So I compute the[br]antiderivative at an endpoint--

0:38:41.789,0:38:44.372
at the other endpoint-- then I'm[br]going to make the difference.

0:38:44.372,0:38:49.700
That's the same thing I do[br]here, so 5 times 3 is 15,

0:38:49.700,0:38:54.410
5 times 0 is 0,[br]15 minus 0 is 15.

0:38:54.410,0:38:56.030
I can keep moving.

0:38:56.030,0:38:59.100
Everything in the[br]parentheses is the number 15.

0:38:59.100,0:39:03.518
I copy and paste, and that[br]should be a piece of cake.

0:39:03.518,0:39:07.011
What do I get?

0:39:07.011,0:39:09.510
STUDENT: 15.

0:39:09.510,0:39:15.560
PROFESSOR: I get 15[br]times x between 0 and 2.

0:39:15.560,0:39:16.910
Integral of 1 is x.

0:39:16.910,0:39:19.780
Integral of 1 is x[br]with respect to x,

0:39:19.780,0:39:24.200
so I get 15 times 2, which[br]is 30, and you go, duh,

0:39:24.200,0:39:26.830
[INAUDIBLE].

0:39:26.830,0:39:28.780
That was elementary mathematics.

0:39:28.780,0:39:32.050
Yes, you were lucky you[br]knew that volume of the box,

0:39:32.050,0:39:35.730
but what if somebody gave[br]you a curvilinear area?

0:39:35.730,0:39:39.360
What if somebody gave you[br]something quite complicated?

0:39:39.360,0:39:40.700
What would you do?

0:39:40.700,0:39:43.453
You have know calculus.

0:39:43.453,0:39:45.918
That's your only chance.

0:39:45.918,0:39:51.720
If you don't calculus,[br]you are dead meat.

0:39:51.720,0:40:00.680
So I'm saying, how[br]about another problem.

0:40:00.680,0:40:04.500
That look like it's[br]complicated, but calculus

0:40:04.500,0:40:08.460
is something[br][INAUDIBLE] with that.

0:40:08.460,0:40:15.755
Suppose that I have a square[br]in the plane between-- this

0:40:15.755,0:40:19.910
is x and y-- do you[br]want square 0,1 0,1

0:40:19.910,0:40:22.630
or you want minus 1[br]to 1 minus 1 to 1.

0:40:22.630,0:40:26.160


0:40:26.160,0:40:28.080
It doesn't matter.

0:40:28.080,0:40:32.970
Well, let's take minus[br]1 to 1 and minus 1 to 1,

0:40:32.970,0:40:35.990
and I'll try to draw[br]as well as I can,

0:40:35.990,0:40:38.182
which I cannot but it's OK.

0:40:38.182,0:40:41.560
You will forgive me.

0:40:41.560,0:40:42.430
This is the floor.

0:40:42.430,0:40:45.615


0:40:45.615,0:40:48.210
If I were just a[br]little tiny square

0:40:48.210,0:40:52.300
in this room plus the[br]equivalent square in that room

0:40:52.300,0:40:53.790
and that room and that room.

0:40:53.790,0:40:56.340
This is the origin.

0:40:56.340,0:40:57.650
Are you guys with me?

0:40:57.650,0:41:00.030
So what you're[br]looking at right now

0:41:00.030,0:41:05.418
is this square foot[br]of carpet that I have,

0:41:05.418,0:41:11.905
but I have another one here and[br]another one behind the wall,

0:41:11.905,0:41:15.590
and so do I everything in mind?

0:41:15.590,0:41:21.130
X is between minus 1 and 1,[br]y is between minus 1 and 1.

0:41:21.130,0:41:24.518


0:41:24.518,0:41:30.350
And somebody gives you z[br]to be a positive function,

0:41:30.350,0:41:36.076
continuous function, which[br]is x squared plus y squared.

0:41:36.076,0:41:37.415
And you go, already.

0:41:37.415,0:41:39.290
Oh, my god.

0:41:39.290,0:41:41.950
I already have this[br]kind of hard function.

0:41:41.950,0:41:44.350
It's not a hard thing to do.

0:41:44.350,0:41:45.380
Let's draw that.

0:41:45.380,0:41:47.990
What are we going to get?

0:41:47.990,0:41:55.980
Your favorite [INAUDIBLE][br]that goes like this.

0:41:55.980,0:41:59.780
And imagine what's[br]going to happen

0:41:59.780,0:42:03.400
with this is like a vase.

0:42:03.400,0:42:06.850
Inside, it has this[br]circular paraboloid.

0:42:06.850,0:42:17.760
But the walls of this vase are--[br]I cannot draw better than that.

0:42:17.760,0:42:25.310
So the walls of this[br]vase are squares.

0:42:25.310,0:42:30.040
And what you have inside is[br]the carved circular paraboloid.

0:42:30.040,0:42:32.580


0:42:32.580,0:42:45.330
Now I'm asking[br]you, how do I find

0:42:45.330,0:43:02.460
volume of the body under and[br]above D, which is minus 1,

0:43:02.460,0:43:03.510
1, minus 1, 1.

0:43:03.510,0:43:05.410
It's hard to draw that, right?

0:43:05.410,0:43:06.970
It's hard to draw.

0:43:06.970,0:43:09.810
So what do we do?

0:43:09.810,0:43:15.507


0:43:15.507,0:43:16.590
We start imagining things.

0:43:16.590,0:43:20.060


0:43:20.060,0:43:24.110
Actually, when you cut with[br]a plane that is y equals 1,

0:43:24.110,0:43:27.870
you would get a parabola.

0:43:27.870,0:43:35.880
And so when you look at what the[br]picture is going to look like,

0:43:35.880,0:43:39.580
you're going to have[br]a parabola like this,

0:43:39.580,0:43:41.790
a parabola like that,[br]exactly the same,

0:43:41.790,0:43:45.990
a parallel parabola like this[br]and a parabola like that.

0:43:45.990,0:43:49.270
Now I started drawing better.

0:43:49.270,0:43:51.581
And you say, how did you[br]start drawing better?

0:43:51.581,0:43:53.370
Well, with a little[br]bit of practice.

0:43:53.370,0:43:59.560
Where are the maxima[br]of this thing?

0:43:59.560,0:44:00.510
At the corners.

0:44:00.510,0:44:01.200
Why is that?

0:44:01.200,0:44:05.650
Because at the corners,[br]you get 1, 1 for both.

0:44:05.650,0:44:10.270
Of course, to do the absolute[br]extrema, minimum, maximum,

0:44:10.270,0:44:14.710
we would have to go back to[br]section 11.7 and do the thing.

0:44:14.710,0:44:19.130
But practically, it's easy[br]to see that at the corners,

0:44:19.130,0:44:23.050
you have the height 2 because[br]this is the point 1, 1.

0:44:23.050,0:44:28.930
And the same height, 2 and 2[br]and 2, are at every corner.

0:44:28.930,0:44:31.870
That would be the[br]maximum that you have.

0:44:31.870,0:44:39.340
So you have 1 minus 1 and so[br]on-- minus 1, 1, and minus 1,

0:44:39.340,0:44:42.389
minus 1, who is behind[br]me, minus 1, minus 1.

0:44:42.389,0:44:46.650
That goes all the way to 2.

0:44:46.650,0:44:51.240
So it's hard to do an[br]approximation with a three

0:44:51.240,0:44:53.090
dimensional model.

0:44:53.090,0:44:54.560
Thank god there is calculus.

0:44:54.560,0:44:59.110
So you say integral of x[br]squared plus y squared,

0:44:59.110,0:45:05.930
as simple as that, da over the[br]domain, D, which is minus 1,

0:45:05.930,0:45:07.820
1, minus 1, 1.

0:45:07.820,0:45:10.370
How do you write it[br]according to the theorem

0:45:10.370,0:45:13.390
that I told you[br]about, Fubini-Tonelli?

0:45:13.390,0:45:19.640
Then you have integral integral[br]x squared plus y squared dy dx.

0:45:19.640,0:45:22.430


0:45:22.430,0:45:25.210
Doesn't matter which[br]one I'm taking.

0:45:25.210,0:45:26.620
I can do dy dx.

0:45:26.620,0:45:27.860
I can do dx dy.

0:45:27.860,0:45:31.220
I just have to pay[br]attention to the endpoints.

0:45:31.220,0:45:33.360
Lucky for you the[br]endpoints are the same.

0:45:33.360,0:45:35.460
y is between minus 1 and 1.

0:45:35.460,0:45:37.443
x is between minus 1 and 1.

0:45:37.443,0:45:40.754


0:45:40.754,0:45:44.720
I wouldn't known how to compute[br]the volume of this vase made

0:45:44.720,0:45:46.094
of marble or made[br]of whatever you

0:45:46.094,0:45:53.835
want to make it unless I knew[br]to compute this integral.

0:45:53.835,0:45:58.582
Now you have to help me[br]because it's not hard

0:45:58.582,0:46:03.902
but it's not easy either, so we[br]need a little bit of attention.

0:46:03.902,0:46:05.860
We always start from the[br]inside to the outside.

0:46:05.860,0:46:10.560
The outer person has to be just[br]neglected for the time being

0:46:10.560,0:46:14.580
and I focus all my attention[br]to this integration.

0:46:14.580,0:46:18.055
And when I integrate[br]with respect to y,

0:46:18.055,0:46:20.450
y is the variable for me.

0:46:20.450,0:46:22.800
Nothing else exists[br]for the time being,

0:46:22.800,0:46:27.540
but y being a variable,[br]x being like a constant.

0:46:27.540,0:46:29.990
So when you integrate x[br]squared plus y squared

0:46:29.990,0:46:34.510
with respect to y, you have[br]to pay attention a little bit.

0:46:34.510,0:46:39.790
It's about the same if you[br]had 7 squared plus y squared.

0:46:39.790,0:46:43.410
So this x squared[br]is like a constant.

0:46:43.410,0:46:45.363
So what do you get inside?

0:46:45.363,0:46:47.362
Let's apply the fundamental[br]theorem of calculus.

0:46:47.362,0:46:48.292
STUDENT: x squared y.

0:46:48.292,0:46:49.250
PROFESSOR: x squared y.

0:46:49.250,0:46:49.960
Excellent.

0:46:49.960,0:46:51.960
I'm very proud of you.

0:46:51.960,0:46:52.670
Plus?

0:46:52.670,0:46:53.800
STUDENT: y cubed over 3.

0:46:53.800,0:46:55.080
PROFESSOR: y cubed over three.

0:46:55.080,0:46:57.460
Again, I'm proud of you.

0:46:57.460,0:47:03.310
Evaluated between y equals[br]minus 1 down, y equals 1 up.

0:47:03.310,0:47:07.020
And I will do the math later[br]because I'm getting tired.

0:47:07.020,0:47:09.730


0:47:09.730,0:47:11.790
Now let's do the math.

0:47:11.790,0:47:13.275
I don't know what[br]I'm going to get.

0:47:13.275,0:47:18.930
I get minus 1 to 1, a[br]big bracket, and dx.

0:47:18.930,0:47:21.930
And in this big bracket, I[br]have to do the difference

0:47:21.930,0:47:23.270
between two values.

0:47:23.270,0:47:26.920
So I put two parentheses.

0:47:26.920,0:47:29.960
When y equals 1, I[br]get x squared 1--

0:47:29.960,0:47:33.800
I'm not going to write[br]that down-- plus 1 cubed

0:47:33.800,0:47:36.510
over 3, 1/3.

0:47:36.510,0:47:41.920
I'm done with evaluating[br]this sausage thingy at 1.

0:47:41.920,0:47:44.310
It's an expression[br]that I evaluate.

0:47:44.310,0:47:46.580
It could be a lot longer.

0:47:46.580,0:47:49.130
I'm not planning to give you[br]long expressions in the midterm

0:47:49.130,0:47:51.936
because you're going to[br]make algebra mistakes,

0:47:51.936,0:47:55.270
and that's not what I want.

0:47:55.270,0:48:01.150
For minus 1, what do we[br]have Minus x squared.

0:48:01.150,0:48:04.500
What is y equals minus[br]1 plugged in here?

0:48:04.500,0:48:05.455
Minus 1/3.

0:48:05.455,0:48:09.092


0:48:09.092,0:48:10.520
I have to pay attention.

0:48:10.520,0:48:15.650
You realize that if I mess[br]up a sign, it's all done.

0:48:15.650,0:48:21.220
So in this case, I say, but[br]this I have minus, minus.

0:48:21.220,0:48:24.182
A minus in front of[br]a minus is a plus,

0:48:24.182,0:48:30.870
so I'm practically doubling[br]the x squared plus 1/3

0:48:30.870,0:48:33.930
and taking it[br]between minus 1 and 1

0:48:33.930,0:48:36.680
and just with respect to x.

0:48:36.680,0:48:38.160
So you say, wait a minute.

0:48:38.160,0:48:38.970
But that's easy.

0:48:38.970,0:48:41.120
I've done that when[br]I was in Calc 1.

0:48:41.120,0:48:41.860
Of course.

0:48:41.860,0:48:47.035
This is the nice part that[br]you get, a simple integral

0:48:47.035,0:48:51.770
from the ones in Calc 1.

0:48:51.770,0:48:56.530
Let's solve this one and find[br]out what the area will be.

0:48:56.530,0:48:59.168
What do we get?

0:48:59.168,0:48:59.980
Is it hard?

0:48:59.980,0:49:00.920
No.

0:49:00.920,0:49:02.150
Kick Mr. 2 out.

0:49:02.150,0:49:04.970
He's just messing[br]up with your life.

0:49:04.970,0:49:06.150
Kick him out.

0:49:06.150,0:49:08.500
2, out.

0:49:08.500,0:49:11.900
And then integral of[br]x squared plus 1/3

0:49:11.900,0:49:15.720
is going to be x[br]cubed over 3 plus--

0:49:15.720,0:49:16.686
STUDENT: x over 3.

0:49:16.686,0:49:18.618
PROFESSOR: x over 3, very good.

0:49:18.618,0:49:22.965
Evaluated between x equals[br]minus 1 down, x equals 1 up.

0:49:22.965,0:49:26.360


0:49:26.360,0:49:27.729
Let's see what we get.

0:49:27.729,0:49:30.942
2 times bracket.

0:49:30.942,0:49:33.565
I'll put a parentheses[br]for the first fractions,

0:49:33.565,0:49:36.980
and another minus, and[br]another parentheses.

0:49:36.980,0:49:42.070
What's the first edition[br]of fractions that I get?

0:49:42.070,0:49:44.440
1/3 plus 1/3.

0:49:44.440,0:49:47.360
I'll put 2/3 because I'm lazy.

0:49:47.360,0:49:49.218
Then minus what?

0:49:49.218,0:49:51.310
STUDENT: Minus 1/3.

0:49:51.310,0:49:56.370
PROFESSOR: Minus 1/3[br]minus 1/3, minus 2/3.

0:49:56.370,0:50:01.120
And now I should be able to[br]not beat around the bush.

0:50:01.120,0:50:04.444
Tell me what the answer[br]will be in the end.

0:50:04.444,0:50:06.270
STUDENT: 8/3.

0:50:06.270,0:50:08.251
PROFESSOR: 8/3.

0:50:08.251,0:50:09.790
Does that make sense?

0:50:09.790,0:50:12.430
When you do that in[br]math, you should always

0:50:12.430,0:50:16.890
think-- one of the famous[br]professors at Harvard

0:50:16.890,0:50:21.650
was saying one time[br]she asked the students,

0:50:21.650,0:50:23.805
how many hours of[br]life do we have have

0:50:23.805,0:50:25.770
in one day, blah, blah, blah?

0:50:25.770,0:50:30.280
And many students[br]came up with 36, 37.

0:50:30.280,0:50:35.560
So always make sure that the[br]answer you get makes sense.

0:50:35.560,0:50:37.890
This is part of a cube, right?

0:50:37.890,0:50:42.920
It's like carved in a[br]cube or a rectangle.

0:50:42.920,0:50:46.490


0:50:46.490,0:50:48.570
Now, what's the height?

0:50:48.570,0:50:53.236
If this were to go[br]up all the way to 2,

0:50:53.236,0:50:58.610
it would be 2, 2, and 2.

0:50:58.610,0:51:04.430
2 times 2 times 2 equals 8,[br]and what we got is 8 over 3.

0:51:04.430,0:51:08.830
Now, using our imagination,[br]it makes sense.

0:51:08.830,0:51:11.380
If I got a 16, I[br]would say, oh my god.

0:51:11.380,0:51:12.220
No, no, no, no.

0:51:12.220,0:51:14.190
What is that?

0:51:14.190,0:51:17.972
So a little bit, I would think,[br]does this make sense or not?

0:51:17.972,0:51:21.710


0:51:21.710,0:51:24.400
Let's do one more,[br]a similar one.

0:51:24.400,0:51:28.180
Now I'm going to count[br]on you a little bit more.

0:51:28.180,0:51:39.246


0:51:39.246,0:51:41.230
STUDENT: Professor,[br]did you calculate that

0:51:41.230,0:51:45.555
by just doing a quarter, and[br]then just multiplying it by 4?

0:51:45.555,0:51:47.096
Because then that[br]would just leave us

0:51:47.096,0:51:48.794
with zeroes [INAUDIBLE].

0:51:48.794,0:51:50.710
PROFESSOR: You mean in[br]that particular figure?

0:51:50.710,0:51:51.209
Yeah.

0:51:51.209,0:51:54.170
STUDENT: Yeah, because it[br]was perfectly [INAUDIBLE].

0:51:54.170,0:51:54.880
PROFESSOR: Yeah.

0:51:54.880,0:51:56.660
It's nice.

0:51:56.660,0:52:02.859
It's a little bit related[br]to some other problems that

0:52:02.859,0:52:03.650
come from pyramids.

0:52:03.650,0:52:06.510


0:52:06.510,0:52:16.246
By the way, how can you compute[br]the volume of a square pyramid?

0:52:16.246,0:52:21.470


0:52:21.470,0:52:26.360
Suppose that you have[br]the same problem.

0:52:26.360,0:52:30.610
Minus 1 to 1 for x and y.

0:52:30.610,0:52:34.760
Minus 1 to 1, minus 1 to 1.

0:52:34.760,0:53:04.560
Let's say the pyramid would[br]have the something like that.

0:53:04.560,0:53:06.646
What would be the volume[br]of such a pyramid?

0:53:06.646,0:53:10.470


0:53:10.470,0:53:12.860
STUDENT: [INAUDIBLE].

0:53:12.860,0:53:17.960
PROFESSOR: The height[br]is h for extra credit.

0:53:17.960,0:53:32.636
Can you compute the[br]volume of this pyramid

0:53:32.636,0:53:33.980
using double integrals?

0:53:33.980,0:53:40.920


0:53:40.920,0:53:48.620
Say the height is h and the[br]bases is the square minus 1,

0:53:48.620,0:53:51.740
1, minus 1, 1.

0:53:51.740,0:53:54.420
I'm sure it can be[br]done, but you know--

0:53:54.420,0:53:58.062
now I'm testing what you[br]remember in terms of geometry

0:53:58.062,0:54:00.691
because we will deal[br]with geometry a lot

0:54:00.691,0:54:03.100
in volumes and areas.

0:54:03.100,0:54:07.475
So how do you do that[br]in general, guys?

0:54:07.475,0:54:09.950
STUDENT: 1/3 [INAUDIBLE].

0:54:09.950,0:54:14.260
PROFESSOR: 1/3 the[br]height times the area

0:54:14.260,0:54:18.536
of the bases, which is what?

0:54:18.536,0:54:20.490
2 times 2.

0:54:20.490,0:54:28.290
2 times 2, 3, over 3, 4/3 h.

0:54:28.290,0:54:30.070
Can you prove that[br]with calculus?

0:54:30.070,0:54:31.110
That's all I'm saying.

0:54:31.110,0:54:33.660
One point extra credit.

0:54:33.660,0:54:36.295
Can you prove that[br]with calculus?

0:54:36.295,0:54:40.610
Actually, you would have[br]to use what you learned.

0:54:40.610,0:54:44.528
You can use Calc 2 as well.

0:54:44.528,0:54:46.710
Do you guys remember[br]that there were

0:54:46.710,0:54:53.270
some cross-sectional areas, like[br]this would be made of cheese,

0:54:53.270,0:54:56.750
and you come with a vertical[br]knife and cut cross sections.

0:54:56.750,0:54:57.910
They go like that.

0:54:57.910,0:54:59.300
But that's awfully hard.

0:54:59.300,0:55:02.995
Maybe you can do it differently[br]with Calc 3 instead of Calc 2.

0:55:02.995,0:55:07.650


0:55:07.650,0:55:10.118
Let's pick one from[br]the book as well.

0:55:10.118,0:55:31.100


0:55:31.100,0:55:33.180
OK.

0:55:33.180,0:55:38.930
So the same idea of using[br]the Fubini-Tonelli argument

0:55:38.930,0:55:45.500
and have an iterative-- evaluate[br]the following double integral

0:55:45.500,0:55:48.820
over the rectangle[br]of vertices 0, 0--

0:55:48.820,0:55:52.120
write it down-- 3,[br]0, 3, 2, and 0, 2.

0:55:52.120,0:56:01.552
So on the bases, you have a[br]rectangle of vertices 3, 0, 0,

0:56:01.552,0:56:14.370
0, 3, 2, and 0, 2.

0:56:14.370,0:56:18.650
And then somebody[br]tells you, find us

0:56:18.650,0:56:29.350
the double integral[br]of 2 minus y da

0:56:29.350,0:56:35.850
over r where r represents the[br]rectangle that we talked about.

0:56:35.850,0:56:37.730
This is exactly [INAUDIBLE].

0:56:37.730,0:56:42.440


0:56:42.440,0:56:45.300
And the answer we[br]should get is 6.

0:56:45.300,0:56:48.920
And I'm saying on top of[br]what we said in the book,

0:56:48.920,0:56:52.980
can you give a geometric[br]interpretation?

0:56:52.980,0:56:55.190
Does this have a[br]geometric interpretation

0:56:55.190,0:56:57.162
you can think of or not?

0:56:57.162,0:57:01.374


0:57:01.374,0:57:04.190
Well, first of all,[br]what is this animal?

0:57:04.190,0:57:07.050
According to the Fubini[br]theorem, this animal

0:57:07.050,0:57:14.160
will have to be-- I have[br]it over a rectangle,

0:57:14.160,0:57:18.110
so assume x will be[br]between a and b, y

0:57:18.110,0:57:22.047
will be between c and d.

0:57:22.047,0:57:24.873
I have to figure[br]out who those are.

0:57:24.873,0:57:31.660
2 minus y and dy dx.

0:57:31.660,0:57:35.675


0:57:35.675,0:57:37.749
Where is y between?

0:57:37.749,0:57:39.540
I should draw the[br]picture for the rectangle

0:57:39.540,0:57:42.760
because otherwise, it's[br]not so easy to see.

0:57:42.760,0:57:50.902
I have 0, 0 here, 3, 0 here, 3,[br]2 over here, shouldn't be hard.

0:57:50.902,0:57:53.220
So this is going to be 0, 2.

0:57:53.220,0:57:57.460
That's the y-axis and[br]that's the x-axis.

0:57:57.460,0:58:00.930
Let's see if we can see it.

0:58:00.930,0:58:05.240
And what is the meaning[br]of the 6, I'm asking you?

0:58:05.240,0:58:07.150
I don't know.

0:58:07.150,0:58:11.380
x should be between[br]0 and 3, right?

0:58:11.380,0:58:15.330
y should be between[br]0 and 2, right?

0:58:15.330,0:58:16.890
Now you are experts in this.

0:58:16.890,0:58:20.916
We've done this twice, and[br]you already know how to do it.

0:58:20.916,0:58:23.160
Integral from 0 to 3.

0:58:23.160,0:58:27.040
Then I take that,[br]and that's going

0:58:27.040,0:58:39.270
to be 2y minus y[br]squared over 2 between y

0:58:39.270,0:58:43.465
equals 0 down and[br]y equals 2 up dx.

0:58:43.465,0:58:48.044


0:58:48.044,0:58:54.690
That means integral from 0[br]to 3, bracket minus bracket

0:58:54.690,0:58:58.690
to make my life easier, dx.

0:58:58.690,0:59:02.450
Now, there is no x, thank god.

0:59:02.450,0:59:04.615
So that means I'm going[br]to have a constant

0:59:04.615,0:59:09.920
minus another constant, which[br]means I go 4 minus 4 over 2.

0:59:09.920,0:59:12.950
2, right?

0:59:12.950,0:59:18.430
The other one, for 0, I get 0.

0:59:18.430,0:59:20.600
I'm very happy I get 0[br]because in that case,

0:59:20.600,0:59:25.170
it's obvious that I get[br]2 times 3, which is 6.

0:59:25.170,0:59:29.410
So I got what the book[br]said I'm going to get.

0:59:29.410,0:59:32.130
But do I have a geometric[br]interpretation of that?

0:59:32.130,0:59:37.160
I would like to see[br]if anybody can--

0:59:37.160,0:59:41.140
I'm going to give you a[br]break in a few minues--

0:59:41.140,0:59:45.970
if anybody can think of a[br]geometric interpretation.

0:59:45.970,0:59:52.630
What is this f of xy if I were[br]to interpret this as a graph?

0:59:52.630,0:59:55.100
x equals f of x and y.

0:59:55.100,0:59:55.760
Is this--

0:59:55.760,0:59:57.520
STUDENT: 2 minus y.

0:59:57.520,1:00:04.550
PROFESSOR: So z equals 2[br]minus y is a plane, right?

1:00:04.550,1:00:08.144
STUDENT: Yes, but then you have[br]the parabola is going down.

1:00:08.144,1:00:11.210
PROFESSOR: And how do I get[br]to draw this plane the best?

1:00:11.210,1:00:13.881
Because there are[br]many ways to do it.

1:00:13.881,1:00:16.700
I look at this wall.

1:00:16.700,1:00:19.120
The y-axis is this.

1:00:19.120,1:00:21.093
The z-axis is the vertical line.

1:00:21.093,1:00:23.406
So I'm looking at this plane.

1:00:23.406,1:00:27.630
y plus z must be equal to 2.

1:00:27.630,1:00:29.850
So when is y plus z equal to 2?

1:00:29.850,1:00:34.150
When I am on a[br]line in the plane.

1:00:34.150,1:00:39.150
I'm going to draw that line[br]with pink because I like pink.

1:00:39.150,1:00:41.240
This is y plus z equals 2.

1:00:41.240,1:00:44.480


1:00:44.480,1:00:50.480
And imagine this line will be[br]shifted by parallelism as it

1:00:50.480,1:00:54.940
comes towards you on all these[br]other parallel vertical planes

1:00:54.940,1:00:57.720
that are parallel to the board.

1:00:57.720,1:01:04.770
So I'm going to have an[br]entire plane like that,

1:01:04.770,1:01:09.300
and I'm going to stop here.

1:01:09.300,1:01:13.105
When I'm in the plane[br]that's called x equals 3--

1:01:13.105,1:01:15.390
this is the plane[br]called x equals

1:01:15.390,1:01:20.160
3-- I have exactly this[br]triangle, this [INAUDIBLE].

1:01:20.160,1:01:23.691
It's in the plane[br]that faces me here.

1:01:23.691,1:01:26.060
I don't know if[br]you realize that.

1:01:26.060,1:01:30.719
I'll help you make a[br]house or something nice.

1:01:30.719,1:01:32.651
I think I'm getting hungry.

1:01:32.651,1:01:35.650
I imagine this again as[br]being a piece of cheese,

1:01:35.650,1:01:39.750
or it looks even like a piece[br]of cake would be with layers.

1:01:39.750,1:01:42.850


1:01:42.850,1:01:47.560
So our question is, if[br]we didn't know calculus

1:01:47.560,1:01:51.210
but we knew how to draw[br]this, and somebody gave you

1:01:51.210,1:01:53.787
this at the GRE[br]or whatever exam,

1:01:53.787,1:01:55.620
how could you have done[br]it without calculus?

1:01:55.620,1:02:00.482
Just by cheating and[br]pretending, I know how to do it,

1:02:00.482,1:02:02.920
but you've never done a[br]double integral in your life.

1:02:02.920,1:02:06.178
So I know it's a volume.

1:02:06.178,1:02:08.655
How do I get the volume?

1:02:08.655,1:02:10.154
What kind of geometric[br]body is that?

1:02:10.154,1:02:11.645
STUDENT: A triangle.

1:02:11.645,1:02:13.633
STUDENT: It's a[br]triangular prism.

1:02:13.633,1:02:15.618
PROFESSOR: It's a[br]triangular prism.

1:02:15.618,1:02:16.118
Good.

1:02:16.118,1:02:19.625
And a triangular prism[br]has what volume formula?

1:02:19.625,1:02:20.750
STUDENT: Base times height.

1:02:20.750,1:02:22.446
PROFESSOR: Base[br]times the height.

1:02:22.446,1:02:25.932
And the height has what area?

1:02:25.932,1:02:27.440
Let's see.

1:02:27.440,1:02:30.218
The base would be that, right?

1:02:30.218,1:02:34.170
And the height would be 3.

1:02:34.170,1:02:36.146
Am I right or not?

1:02:36.146,1:02:37.581
The height would be 3.

1:02:37.581,1:02:38.122
This is not--

1:02:38.122,1:02:38.990
STUDENT: It's 2.

1:02:38.990,1:02:39.490
Yeah.

1:02:39.490,1:02:40.324
STUDENT: No, it's 3.

1:02:40.324,1:02:41.906
DR. MAGDALENA TODA:[br]From here to here?

1:02:41.906,1:02:42.450
STUDENT: 3.

1:02:42.450,1:02:43.575
DR. MAGDALENA TODA: It's 3.

1:02:43.575,1:02:46.860
So how much is that?

1:02:46.860,1:02:47.690
How much-- OK.

1:02:47.690,1:02:50.010
From here to here is 2.

1:02:50.010,1:02:54.200
From here to here,[br]it's how much?

1:02:54.200,1:02:56.119
STUDENT: The height[br]is only-- I see--

1:02:56.119,1:02:57.065
STUDENT: It's also 2.

1:02:57.065,1:02:59.440
DR. MAGDALENA TODA: It's[br]also 2 because look at that.

1:02:59.440,1:03:01.825
It's an isosceles triangle.

1:03:01.825,1:03:03.730
This is 45 to 45.

1:03:03.730,1:03:05.250
So this is also 2.

1:03:05.250,1:03:08.860
2 to-- that's 90[br]degrees, 45, 45.

1:03:08.860,1:03:09.360
OK.

1:03:09.360,1:03:13.220
So the area of the shaded purple[br]triangle-- how much is that?

1:03:13.220,1:03:13.964
STUDENT: 2.

1:03:13.964,1:03:14.880
DR. MAGDALENA TODA: 2.

1:03:14.880,1:03:17.120
2 times 2 over 2.

1:03:17.120,1:03:19.710
2 times 3 equals 6.

1:03:19.710,1:03:22.220
I don't need calculus.

1:03:22.220,1:03:24.170
In this case, I[br]don't need calculus.

1:03:24.170,1:03:27.170
But when I have those[br]nasty curvilinear

1:03:27.170,1:03:31.958
z equals f of x, y, complicated[br]expressions, I have no choice.

1:03:31.958,1:03:34.946
I have to do the[br]double integral.

1:03:34.946,1:03:37.950
But in this case, even if[br]I didn't know how to do it,

1:03:37.950,1:03:39.200
I would still get the 6.

1:03:39.200,1:03:39.976
Yes, sir?

1:03:39.976,1:03:42.738
STUDENT: What if we[br]did that on the exam?

1:03:42.738,1:03:44.321
DR. MAGDALENA TODA:[br]Well, that's good.

1:03:44.321,1:03:45.736
I will then keep it in mind.

1:03:45.736,1:03:46.235
Yes.

1:03:46.235,1:03:48.560
It doesn't matter to me.

1:03:48.560,1:03:50.900
I have other colleagues who[br]really care about the method

1:03:50.900,1:03:52.350
and start complaining.

1:03:52.350,1:03:55.560
I don't care how you[br]get to the answer

1:03:55.560,1:03:57.450
as long as you got[br]the right answer.

1:03:57.450,1:03:59.400
Let me tell you my logic.

1:03:59.400,1:04:03.830
Suppose somebody hired you[br]thinking you're a good worker,

1:04:03.830,1:04:05.327
and you're smart and so on.

1:04:05.327,1:04:09.818
Would they care how you got to[br]the solution of the problem?

1:04:09.818,1:04:14.220
As long as the problem[br]was solved correctly, no.

1:04:14.220,1:04:18.250
And actually, the elementary[br]way is the fastest

1:04:18.250,1:04:20.120
because it's just 10 seconds.

1:04:20.120,1:04:20.923
You draw.

1:04:20.923,1:04:21.690
You imagine.

1:04:21.690,1:04:23.180
You know what it is.

1:04:23.180,1:04:28.310
So your boss will want you to[br]find the fastest way to provide

1:04:28.310,1:04:29.230
the correct solution.

1:04:29.230,1:04:33.480
He's not going to[br]care how you got that.

1:04:33.480,1:04:35.580
So no matter how[br]you do it, as long

1:04:35.580,1:04:39.990
as you've got the right[br]answer, I'm going to be happy.

1:04:39.990,1:04:49.230
I want to ask you to please[br]go to page 927 in the book

1:04:49.230,1:04:50.380
and read.

1:04:50.380,1:04:52.580
It's only one page.

1:04:52.580,1:04:55.340
That whole end section, 12.1.

1:04:55.340,1:04:59.928
It's called an informal[br]argument for Fubini's theorem.

1:04:59.928,1:05:05.736
Practically, it's a proof of[br]Fubini's theorem, page 927.

1:05:05.736,1:05:08.640
And then I'm going to go[br]ahead and start the homework

1:05:08.640,1:05:11.820
four, if you don't mind.

1:05:11.820,1:05:15.920
I'm going to go into WeBWork[br]and give you homework four.

1:05:15.920,1:05:18.580
And the first few[br]problems that you

1:05:18.580,1:05:21.100
are going to be[br]expected to solve

1:05:21.100,1:05:27.313
will be out of 12.1,[br]which is really easy.

1:05:27.313,1:05:28.792
I'll give you a[br]few minutes back.

1:05:28.792,1:05:32.736
And we go on with 12.2,[br]and it's very similar.

1:05:32.736,1:05:34.708
You're going to like that.

1:05:34.708,1:05:39.638
And then we'll go home or[br]wherever we need to go.

1:05:39.638,1:05:42.103
So you have a few[br]minutes of a break.

1:05:42.103,1:05:45.560
Pick up your extra credits.

1:05:45.560,1:05:47.129
I'll call the names.

1:05:47.129,1:05:48.596
Lily.

1:05:48.596,1:05:52.019
You got a lot of points.

1:05:52.019,1:05:54.953
And [INAUDIBLE].

1:05:54.953,1:05:56.909
And you have two separate ones.

1:05:56.909,1:05:58.365
Nathan.

1:05:58.365,1:05:58.865
Nathan?

1:05:58.865,1:06:02.310


1:06:02.310,1:06:03.040
Rachel Smith.

1:06:03.040,1:06:05.730


1:06:05.730,1:06:06.230
Austin.

1:06:06.230,1:06:09.278


1:06:09.278,1:06:09.778
Thank you.

1:06:09.778,1:06:12.736


1:06:12.736,1:06:13.722
Edgar.

1:06:13.722,1:06:16.180
[INAUDIBLE]

1:06:16.180,1:06:16.680
Aaron.

1:06:16.680,1:06:24.068


1:06:24.068,1:06:24.568
Andre.

1:06:24.568,1:06:32.456


1:06:32.456,1:06:35.407
Aaron.

1:06:35.407,1:06:35.907
Kasey.

1:06:35.907,1:06:39.851


1:06:39.851,1:06:43.490
Kasey came up with[br]a very good idea

1:06:43.490,1:06:47.530
that I will write[br]a review sample.

1:06:47.530,1:06:48.615
Did I promise that?

1:06:48.615,1:06:52.200
A review sample for the midterm.

1:06:52.200,1:06:53.720
And so I said yes.

1:06:53.720,1:06:56.696


1:06:56.696,1:07:01.160
Karen and Matthew.

1:07:01.160,1:07:07.604


1:07:07.604,1:07:08.104
Reagan.

1:07:08.104,1:07:16.040


1:07:16.040,1:07:17.910
Aaron.

1:07:17.910,1:07:20.510
When you submitted,[br]you submitted.

1:07:20.510,1:07:21.194
Yeah.

1:07:21.194,1:07:21.860
And [INAUDIBLE].

1:07:21.860,1:07:25.860


1:07:25.860,1:07:26.860
here.

1:07:26.860,1:07:27.860
And I'm done.

1:07:27.860,1:07:46.360


1:07:46.360,1:07:48.360
STUDENT: Did we[br]turn in [INAUDIBLE]?

1:07:48.360,1:07:49.860
DR. MAGDALENA TODA:[br]Yes, absolutely.

1:07:49.860,1:08:08.860


1:08:08.860,1:08:12.438
Now once we go over[br]12.2, you will say, oh,

1:08:12.438,1:08:14.434
but I understand[br]the Fubini theorem.

1:08:14.434,1:08:21.439


1:08:21.439,1:08:23.926
I didn't know whether[br]there's room for Fubini,

1:08:23.926,1:08:29.250
because once I cover the more[br]general case, which is in 12.2,

1:08:29.250,1:08:33.580
you are going to understand[br]Why Fubini-Tonelli

1:08:33.580,1:08:36.970
works for rectangles.

1:08:36.970,1:08:47.220
So if I think of a domain[br]that is of the following form,

1:08:47.220,1:08:54.207
in the x, y plane, I go x[br]is between and and b, right?

1:08:54.207,1:08:59.685
That's my favorite x.

1:08:59.685,1:09:02.340
So I take the pink[br]segment, and I

1:09:02.340,1:09:05.050
say, everything that[br]happens-- it's going

1:09:05.050,1:09:08.450
to happen on top of this world.

1:09:08.450,1:09:11.359
I have, let's say,[br]two functions.

1:09:11.359,1:09:14.100
To make my life easier, I'll[br]assume both of them [INAUDIBLE]

1:09:14.100,1:09:15.830
one bigger than the other.

1:09:15.830,1:09:23.880
But in case they are[br]not both positive,

1:09:23.880,1:09:28.080
I just need f to be bigger[br]than g for every point.

1:09:28.080,1:09:32.742
And the same argument[br]will function.

1:09:32.742,1:09:38.890
This is f, continuous positive.

1:09:38.890,1:09:41.996
Then g, continuous[br]positive but smaller

1:09:41.996,1:09:44.912
in values than this one.

1:09:44.912,1:09:47.828


1:09:47.828,1:09:48.800
Yes, sir?

1:09:48.800,1:09:51.189
STUDENT: [INAUDIBLE][br]12.2 that we're starting?

1:09:51.189,1:09:52.229
DR. MAGDALENA TODA: 12.2.

1:09:52.229,1:09:56.005
And you are more organized[br]than I am, and I appreciate it.

1:09:56.005,1:10:02.453
So integration over a[br]non-rectangular domain.

1:10:02.453,1:10:06.930


1:10:06.930,1:10:10.361
And we call this a[br]type one because this

1:10:10.361,1:10:12.325
is what many books are using.

1:10:12.325,1:10:17.030
And this is that x is[br]between two fixed end points.

1:10:17.030,1:10:21.270
But y is between two[br]variable end points.

1:10:21.270,1:10:24.300
So what's going to happen to y?

1:10:24.300,1:10:29.480
y is going to take[br]values between the lower,

1:10:29.480,1:10:34.752
the bottom one, which is[br]g of x, and the upper one,

1:10:34.752,1:10:37.062
which is f of x.

1:10:37.062,1:10:39.730
So this is how we[br]define the domain that's

1:10:39.730,1:10:45.230
shaded by me with black[br]shades, vertical strips here.

1:10:45.230,1:10:47.980
This is the domain.

1:10:47.980,1:10:56.240
Now you really do[br]not need to prove

1:10:56.240,1:11:09.250
that double integral over[br]1 dA over-- let's call

1:11:09.250,1:11:15.420
the domain D-- is what?

1:11:15.420,1:11:17.980


1:11:17.980,1:11:27.776
Integral between f of x[br]minus g of x from a to b dx.

1:11:27.776,1:11:31.080


1:11:31.080,1:11:32.087
And you say, what?

1:11:32.087,1:11:33.670
Magdalena, what are[br]you trying to say?

1:11:33.670,1:11:34.760
OK.

1:11:34.760,1:11:37.330
Let's go back and[br]say, what if somebody

1:11:37.330,1:11:41.370
would have asked you the[br]same question in calculus 2?

1:11:41.370,1:11:44.640
Saying, guys I have a[br]question about the area

1:11:44.640,1:11:49.080
in the shaded strip,[br]vertical strip thing.

1:11:49.080,1:11:50.869
How are we going[br]to compute that?

1:11:50.869,1:11:53.590
And you would say,[br]oh, I have an idea.

1:11:53.590,1:12:04.020
I take the area under the graph[br]f, and I shade that in orange.

1:12:04.020,1:12:05.917
And I know what that is.

1:12:05.917,1:12:07.500
So you would say, I[br]know what that is.

1:12:07.500,1:12:08.990
That's going to be what?

1:12:08.990,1:12:13.276
Integral from a to be f of x dx.

1:12:13.276,1:12:16.890
Let's call that A1, right?

1:12:16.890,1:12:19.631
A1.

1:12:19.631,1:12:27.920
Then you go, minus the area[br]with-- I'm just going to shade

1:12:27.920,1:12:32.128
that, brown strips under g.

1:12:32.128,1:12:35.050


1:12:35.050,1:12:37.980
g of x dx.

1:12:37.980,1:12:39.320
And call that A2.

1:12:39.320,1:12:42.310


1:12:42.310,1:12:45.300
A1 minus A2.

1:12:45.300,1:12:49.330
We know both of these[br]formulas from where?

1:12:49.330,1:12:52.900
Calc 1 because that's where[br]you learned about the area

1:12:52.900,1:12:55.040
under the graph of a curve.

1:12:55.040,1:12:57.950
This is the area under[br]the graph of a curve f.

1:12:57.950,1:13:00.880
This is the area under[br]the graph of the curve g.

1:13:00.880,1:13:04.550
The black striped area[br]is their difference.

1:13:04.550,1:13:05.160
All right.

1:13:05.160,1:13:07.190
And so how much is that?

1:13:07.190,1:13:09.130
I'm sorry I put the wrong thing.

1:13:09.130,1:13:11.560
a, b.

1:13:11.560,1:13:13.530
That's going to be[br]integral from a to b.

1:13:13.530,1:13:15.890
Now you say, wait,[br]wait, wait a minute.

1:13:15.890,1:13:17.126
Based on what?

1:13:17.126,1:13:20.260
Based on some sort of[br]additivity property

1:13:20.260,1:13:23.640
of the integral of one[br]variable, which says integral

1:13:23.640,1:13:27.025
from a to b of f plus g.

1:13:27.025,1:13:29.270
You can have f plus, minus g.

1:13:29.270,1:13:30.540
It doesn't matter.

1:13:30.540,1:13:31.940
dx.

1:13:31.940,1:13:37.882
You have integral from a to b f[br]dx plus integral from a to b g

1:13:37.882,1:13:39.176
dx.

1:13:39.176,1:13:42.390
It doesn't matter what.

1:13:42.390,1:13:46.050
You can have a linear[br]combination of f and g.

1:13:46.050,1:13:46.910
Yes, Matthew?

1:13:46.910,1:13:49.180
MATTHEW: So this is[br]just for the domain?

1:13:49.180,1:13:52.790
So if you put it,[br]that would be down.

1:13:52.790,1:13:55.530
So there might be[br]another formula up here

1:13:55.530,1:13:57.350
that would be curved surface.

1:13:57.350,1:13:59.560
And this is the bottom,[br]so you're using integral

1:13:59.560,1:14:01.260
to find the base,[br]and then you're

1:14:01.260,1:14:03.706
going to plug that integral[br]into the other integral.

1:14:03.706,1:14:06.080
DR. MAGDALENA TODA: So I'm[br]just using the property that's

1:14:06.080,1:14:10.660
called linearity of[br]the simple integral,

1:14:10.660,1:14:14.660
meaning that if I have even[br]a linear combination like af

1:14:14.660,1:14:22.246
plus bg, then a-- I have not a.

1:14:22.246,1:14:26.758
Let me call it big A and[br]big B. Big A Af integral

1:14:26.758,1:14:29.360
of f plus big B integral of g.

1:14:29.360,1:14:30.610
You've learned that in Calc 2.

1:14:30.610,1:14:34.390
I'm doing this to apply it for[br]these areas that are subtracted

1:14:34.390,1:14:36.250
from one another.

1:14:36.250,1:14:39.120
If I were to add, as you[br]said, I would put something

1:14:39.120,1:14:39.990
on top of that.

1:14:39.990,1:14:44.650
And then it would be like[br]a superimposition onto it.

1:14:44.650,1:14:54.190
So I have integral from a to[br]b of f of x minus g of x dx.

1:14:54.190,1:14:56.840
And I claim that[br]this is the same

1:14:56.840,1:15:06.900
as double integral of the[br]1dA over the domain D.

1:15:06.900,1:15:10.166
How can you write[br]that differently?

1:15:10.166,1:15:12.040
I'll tell you how you[br]write that differently.

1:15:12.040,1:15:19.150
Integral from a to b of[br]integral from-- what's

1:15:19.150,1:15:21.340
the bottom value of Mr. Y?

1:15:21.340,1:15:23.880


1:15:23.880,1:15:26.620
So Mr. X knows what he's doing.

1:15:26.620,1:15:28.620
He goes all the way from a to b.

1:15:28.620,1:15:31.330
The bottom value of y is g of x.

1:15:31.330,1:15:36.250
You go from the bottom value[br]of y g of x to the upper value

1:15:36.250,1:15:38.080
f of x.

1:15:38.080,1:15:42.320
And then you here put 1 and dy.

1:15:42.320,1:15:44.660
Is this the same thing?

1:15:44.660,1:15:46.440
You say, OK, I know this one.

1:15:46.440,1:15:49.290
I know this one from calc 2.

1:15:49.290,1:15:53.890
But Magdalena, the one[br]you gave us is new.

1:15:53.890,1:15:55.450
It's new and not new, guys.

1:15:55.450,1:15:59.360
This is Fubini's[br]theorem but generalized

1:15:59.360,1:16:01.460
to something that depends on x.

1:16:01.460,1:16:02.890
So how do I do that?

1:16:02.890,1:16:05.080
Integral of 1dy.

1:16:05.080,1:16:07.228
That's what?

1:16:07.228,1:16:11.988
That's y measured between two[br]values that don't depend on y.

1:16:11.988,1:16:16.670
They depend only on x, g of x on[br]the bottom, f of x on the top.

1:16:16.670,1:16:20.155
So this is exactly the[br]integral from a to b.

1:16:20.155,1:16:22.360
In terms of the[br]round parentheses,

1:16:22.360,1:16:25.830
I put-- what is y between[br]f of x and g of x?

1:16:25.830,1:16:30.000
f of x minus g of x dx.

1:16:30.000,1:16:34.474
So it is exactly the[br]same thing from Calc 2

1:16:34.474,1:16:36.410
expressed as a double integral.

1:16:36.410,1:16:42.220


1:16:42.220,1:16:42.920
All right.

1:16:42.920,1:16:54.156
Now This is a type one[br]region that we talked about.

1:16:54.156,1:16:59.600
A type two region is a[br]similar region, practically.

1:16:59.600,1:17:03.130
What you have to keep[br]in mind is they're both

1:17:03.130,1:17:05.770
given here as examples.

1:17:05.770,1:17:09.255
But the technique is[br]absolutely the same.

1:17:09.255,1:17:13.100
If instead of[br]taking this picture,

1:17:13.100,1:17:20.012
I would take y to move[br]between fixed values,

1:17:20.012,1:17:26.278
like y has to be between[br]c and d-- this is my y.

1:17:26.278,1:17:28.770
These are the fixed values.

1:17:28.770,1:17:33.240
And then give me[br]some nice colors.

1:17:33.240,1:17:42.180
This curve and[br]that curve-- OK, I

1:17:42.180,1:17:49.860
have to rotate my head because[br]then this is going to be x.

1:17:49.860,1:17:51.780
This is going to be y.

1:17:51.780,1:17:57.000
And the blue thingy has[br]to be a function of y.

1:17:57.000,1:17:58.810
x is a function of y.

1:17:58.810,1:18:01.338
So how do I call that?

1:18:01.338,1:18:09.950
I have x or whatever[br]equals big F of y.

1:18:09.950,1:18:16.930
And here in the red one, I[br]have x equals big G of y.

1:18:16.930,1:18:23.280
And how am I going to[br]evaluate the striped area?

1:18:23.280,1:18:30.550
Of course striped because I[br]have again y is between c and d.

1:18:30.550,1:18:33.770
And what's moving is Mr. X.

1:18:33.770,1:18:37.480
And Mr. X refuses to[br]have fixed variables.

1:18:37.480,1:18:41.961
Now he goes, I move from[br]the bottom, which is G of y,

1:18:41.961,1:18:46.871
to the top, which is F of y.

1:18:46.871,1:18:50.800
How am I going to write[br]the double integral

1:18:50.800,1:18:58.236
over this domain of[br]1dA, where dA is dxdy.

1:18:58.236,1:19:00.380
Who's going to tell me?

1:19:00.380,1:19:05.140
Similarly, the same[br]reasoning as for this one.

1:19:05.140,1:19:10.310
I'm going to have the[br]integral from what to what

1:19:10.310,1:19:12.190
of integral from what to what?

1:19:12.190,1:19:14.840
Who comes first, dx or dy?

1:19:14.840,1:19:15.524
STUDENT: dx.

1:19:15.524,1:19:16.940
DR. MAGDALENA TODA:[br]dx, very good.

1:19:16.940,1:19:18.610
And dy at the end.

1:19:18.610,1:19:23.130
So y will be between[br]c and d, and x

1:19:23.130,1:19:31.530
is going to be between[br]G of y and F of y.

1:19:31.530,1:19:32.440
And here is y.

1:19:32.440,1:19:35.310


1:19:35.310,1:19:38.510
How can I rewrite this integral?

1:19:38.510,1:19:40.010
Very easily.

1:19:40.010,1:19:46.070
The integral from c to[br]d of the guy on top,

1:19:46.070,1:19:54.090
the blue guy, F of y, minus the[br]guy on the bottom, G of y, dy.

1:19:54.090,1:20:00.170
Some people call the[br]vertical stip method

1:20:00.170,1:20:02.910
compared to the horizontal[br]strip method, where

1:20:02.910,1:20:05.150
in this kind of[br]horizontal strip method,

1:20:05.150,1:20:08.020
you just have to view[br]x as a function of y

1:20:08.020,1:20:11.730
and rotate your head and apply[br]the same reasoning as before.

1:20:11.730,1:20:13.140
It's not a big deal.

1:20:13.140,1:20:15.900
You just need a little[br]bit of imagination,

1:20:15.900,1:20:20.464
and the result is the same.

1:20:20.464,1:20:24.622
An example that's[br]not too hard-- I

1:20:24.622,1:20:26.452
want to give you[br]several examples.

1:20:26.452,1:20:29.450


1:20:29.450,1:20:31.326
We have plenty of time.

1:20:31.326,1:20:36.050
Now it says, we have[br]a triangular region.

1:20:36.050,1:20:40.980
And that is enclosed by lines[br]y equals 0, y equals 2x,

1:20:40.980,1:20:43.730
and x equals 1.

1:20:43.730,1:20:47.520
Let's see what that means[br]and be able to draw it.

1:20:47.520,1:20:51.390
It's very important to be[br]able to draw in this chapter.

1:20:51.390,1:20:54.770
If you're not, just[br]learn how to draw,

1:20:54.770,1:20:56.690
and that will give[br]you lots of ideas

1:20:56.690,1:20:58.392
on how to solve the problems.

1:20:58.392,1:21:18.270


1:21:18.270,1:21:22.878
Chapter 12 is included[br]completely on the midterm.

1:21:22.878,1:21:25.368
So the midterm is[br]on the 2nd of April.

1:21:25.368,1:21:29.850
For the midterm, we have chapter[br]10, those three sections.

1:21:29.850,1:21:32.340
Then we have chapter[br]11 completely,

1:21:32.340,1:21:40.190
and then we have chapter 12[br]not completely, up to 12.6.

1:21:40.190,1:21:40.760
All right.

1:21:40.760,1:21:44.400
So what did I say?

1:21:44.400,1:21:49.140
I have a triangular region that[br]is obtained by intersecting

1:21:49.140,1:21:50.900
the following lines.

1:21:50.900,1:21:58.730
y equals 0, x equals[br]1, and y equals 2x.

1:21:58.730,1:22:01.938
Can I draw them and[br]see how they intersect?

1:22:01.938,1:22:03.340
It shouldn't be a big problem.

1:22:03.340,1:22:05.850
This is a line that[br]passes through the origin

1:22:05.850,1:22:07.990
and has slope 2.

1:22:07.990,1:22:10.980
So it should be[br]very easy to draw.

1:22:10.980,1:22:18.270
At 1, x equals 1, the y will[br]be 2 for this line of slope 2.

1:22:18.270,1:22:20.850
So I'll try to draw.

1:22:20.850,1:22:23.720
Does this look double to you?

1:22:23.720,1:22:29.260
So this is 2.

1:22:29.260,1:22:32.152
This is the point 1, 2.

1:22:32.152,1:22:35.220
And that's the line y equals 2x.

1:22:35.220,1:22:38.410
And that's the line y equals 0.

1:22:38.410,1:22:40.220
And that's the line x equals 1.

1:22:40.220,1:22:43.450
So can I shade this triangle?

1:22:43.450,1:22:47.560
Yeah, I can eventually,[br]depending on what they ask me.

1:22:47.560,1:22:49.175
What do they ask me?

1:22:49.175,1:22:58.100
Find the double[br]integral of x plus y dA

1:22:58.100,1:23:05.600
with respect to the area element[br]over T, T being the triangle.

1:23:05.600,1:23:09.508
So now I'm going to ask,[br]did they say by what method?

1:23:09.508,1:23:12.650
Unfortunately, they say,[br]do it by both methods.

1:23:12.650,1:23:17.190
That means both by x[br]intregration first and then

1:23:17.190,1:23:20.090
y integration and[br]the other way around.

1:23:20.090,1:23:23.310
So they ask you to change[br]the order of the integration

1:23:23.310,1:23:24.720
or do what?

1:23:24.720,1:23:27.270
Switch from vertical[br]strip method

1:23:27.270,1:23:29.600
to horizontal strip method.

1:23:29.600,1:23:31.150
You should get the same answer.

1:23:31.150,1:23:34.160
That's a typical[br]final exam problem.

1:23:34.160,1:23:40.410
When we test you, if[br]you are able to do this

1:23:40.410,1:23:43.314
through the vertical[br]strip or horizontal

1:23:43.314,1:23:45.250
strip and change the[br]order of integration.

1:23:45.250,1:23:47.560
If I do it with the[br]vertical strip method,

1:23:47.560,1:23:52.010
who comes first,[br]the dy or the dx?

1:23:52.010,1:23:53.310
Think a little bit.

1:23:53.310,1:23:55.610
Where do I put d--[br]Fubini [INAUDIBLE]

1:23:55.610,1:23:58.652
comes dy dx or dx dy?

1:23:58.652,1:23:59.600
STUDENT: dy.

1:23:59.600,1:24:01.550
PROFESSOR: dy dx.

1:24:01.550,1:24:04.360
So VSM.

1:24:04.360,1:24:06.560
You're going to laugh.

1:24:06.560,1:24:07.890
It's not written in the book.

1:24:07.890,1:24:10.810
It's like a childish name,[br]Vertical Strip Method,

1:24:10.810,1:24:12.615
meeting integration[br]with respect to y

1:24:12.615,1:24:14.924
and then with respect to x.

1:24:14.924,1:24:17.726
It helped my students[br]through the last decade

1:24:17.726,1:24:19.594
to remember about[br]the vertical strips.

1:24:19.594,1:24:25.490
And that's why I say something[br]that's not using the book, VSM.

1:24:25.490,1:24:35.810
Now, I have integral from-- so[br]who is Mr. X going from 0 to 1?

1:24:35.810,1:24:36.345
He's stable.

1:24:36.345,1:24:37.750
He's happy.

1:24:37.750,1:24:39.700
He's going between[br]two fixed values.

1:24:39.700,1:24:43.770
y goes between the[br]bottom line, which is 0.

1:24:43.770,1:24:44.650
We are lucky.

1:24:44.650,1:24:47.600
It's a really nice problem.

1:24:47.600,1:24:51.396
Going to y equals 2x.

1:24:51.396,1:24:54.020
So it's not hard at all.

1:24:54.020,1:24:59.230
And we have to integrate[br]the function x plus y.

1:24:59.230,1:25:01.768
It should be a piece of cake.

1:25:01.768,1:25:06.600
Let's do this together because[br]you've accumulated seniority

1:25:06.600,1:25:07.675
in this type of problem.

1:25:07.675,1:25:10.520


1:25:10.520,1:25:12.440
What do I put inside?

1:25:12.440,1:25:14.580
What's integral of x[br]plus y with respect to y?

1:25:14.580,1:25:15.987
Is it hard?

1:25:15.987,1:25:19.270


1:25:19.270,1:25:23.446
xy plus-- somebody tell me.

1:25:23.446,1:25:25.110
STUDENT: y squared.

1:25:25.110,1:25:29.160
PROFESSOR: y squared[br]over 2, between y

1:25:29.160,1:25:33.450
equals 0 on the bottom,[br]y equals 2x on top.

1:25:33.450,1:25:36.860
I have to be smart and[br]plug in the values y.

1:25:36.860,1:25:39.009
Otherwise, I'll never make it.

1:25:39.009,1:25:39.800
STUDENT: Professor?

1:25:39.800,1:25:40.900
PROFESSOR: Yes, sir?

1:25:40.900,1:25:43.310
STUDENT: Why did you take[br]2x as the final value

1:25:43.310,1:25:45.310
because you have a[br]specified triangle.

1:25:45.310,1:25:48.970
PROFESSOR: Because y[br]equals 2x is the expression

1:25:48.970,1:25:52.100
of the upper function.

1:25:52.100,1:25:54.775
The upper function is[br]the line y equals 2x.

1:25:54.775,1:25:56.190
They provided that.

1:25:56.190,1:25:59.660
So from the bottom function[br]to the upper function,

1:25:59.660,1:26:02.376
the vertical strips go[br]between two functions.

1:26:02.376,1:26:05.280


1:26:05.280,1:26:07.720
So when I plug in[br]here y equals 2x,

1:26:07.720,1:26:09.723
I have to pay attention[br]to my algebra.

1:26:09.723,1:26:13.820
If I forget the 2, it's all[br]over for me, zero points.

1:26:13.820,1:26:16.260
Well, not zero points,[br]but 10% credit.

1:26:16.260,1:26:20.090
I have no idea what I would[br]get, so I have to pay attention.

1:26:20.090,1:26:26.616
2x times x is 2x squared[br]plus 2x all squared-- guys,

1:26:26.616,1:26:30.520
keep an eye on me--[br]4x squared over 2.

1:26:30.520,1:26:36.430
I put the first value[br]in a pink parentheses,

1:26:36.430,1:26:40.804
and then I move on to[br]the line parentheses.

1:26:40.804,1:26:42.840
Evaluate it at 0.

1:26:42.840,1:26:44.506
That line is very lucky.

1:26:44.506,1:26:50.910
I get a 0 because y[br]equals 0 will give me 0.

1:26:50.910,1:26:54.180
What am I going to get here?

1:26:54.180,1:26:56.521
2x squared plus 2x squared.

1:26:56.521,1:26:57.021
Good.

1:26:57.021,1:26:59.406
What's 2x squared[br]plus 2x squared?

1:26:59.406,1:27:00.260
4x squared.

1:27:00.260,1:27:01.916
So a 4 goes out.

1:27:01.916,1:27:03.320
Kick him out.

1:27:03.320,1:27:06.190
Integral from 0[br]to 1 x squared dx.

1:27:06.190,1:27:08.050
Integral of x squared is?

1:27:08.050,1:27:11.650


1:27:11.650,1:27:14.100
Integral of x squared is?

1:27:14.100,1:27:15.100
STUDENT: x cubed over 3.

1:27:15.100,1:27:16.183
PROFESSOR: x cubed over 3.

1:27:16.183,1:27:19.332
And if you take it[br]between 1 and 0, you get?

1:27:19.332,1:27:20.660
STUDENT: 1.

1:27:20.660,1:27:21.310
PROFESSOR: 1/3.

1:27:21.310,1:27:23.700
1/3 times 4 is 4/3.

1:27:23.700,1:27:26.820


1:27:26.820,1:27:29.140
Suppose this is going to[br]happen on the midterm,

1:27:29.140,1:27:32.400
and I'm asking you to do it[br]reversing the integration

1:27:32.400,1:27:33.930
order.

1:27:33.930,1:27:37.520
Then you are going to check[br]your own work very beautifully

1:27:37.520,1:27:41.720
in the sense that[br]you say, well, now

1:27:41.720,1:27:45.880
I'm going to see if I made[br]a mistake in this one.

1:27:45.880,1:27:46.690
What do I do?

1:27:46.690,1:27:50.480
I erase the whole thing, and[br]instead of vertical strips,

1:27:50.480,1:27:55.840
I'm going to put[br]horizontal strips.

1:27:55.840,1:28:01.020
And you say, well, life is a[br]little bit harder in this case

1:28:01.020,1:28:04.526
because in this[br]case, I have to look

1:28:04.526,1:28:10.850
at y between fixed[br]values, y between 0 and 1.

1:28:10.850,1:28:17.785
So y is between 0 and 1--[br]0 and 2, fixed values.

1:28:17.785,1:28:22.840
And Mr. X says, I'm going[br]between two functions of y.

1:28:22.840,1:28:26.230
I don't know what those[br]functions of y are.

1:28:26.230,1:28:28.380
I'm puzzled.

1:28:28.380,1:28:30.600
You have to help[br]Mr. X know where

1:28:30.600,1:28:34.520
he's going because his life[br]right now is a little bit hard.

1:28:34.520,1:28:39.295
So what is the[br]function for the blue?

1:28:39.295,1:28:42.205


1:28:42.205,1:28:44.160
Now he's not blue anymore.

1:28:44.160,1:28:45.140
He's brown.

1:28:45.140,1:28:47.840
x equals 1.

1:28:47.840,1:28:50.100
So he knows what[br]he's going to be.

1:28:50.100,1:28:52.812
What is the x function[br]for the red line

1:28:52.812,1:28:54.580
that [INAUDIBLE] asked about?

1:28:54.580,1:28:55.464
STUDENT: y over 2.

1:28:55.464,1:28:57.550
PROFESSOR: x must be y over 2.

1:28:57.550,1:29:01.010
It's the same thing, but I have[br]to express x in terms of y.

1:29:01.010,1:29:05.282
So I erase and I say[br]x equals y over 2.

1:29:05.282,1:29:06.700
Same thing.

1:29:06.700,1:29:11.140
So x has to be between what and[br]what, the bottom and the top?

1:29:11.140,1:29:13.680
Well, I turn my head.

1:29:13.680,1:29:19.391
The top must be x equals 1,[br]and the bottom one is y over 2.

1:29:19.391,1:29:24.980
That's the bottom one,[br]the bottom value for x.

1:29:24.980,1:29:27.450
Now wish me luck because I[br]have to get the same thing.

1:29:27.450,1:29:35.520
So integral from 0 to 2 of[br]integral from y over 2 to 1.

1:29:35.520,1:29:37.512
Changing the order[br]of integration

1:29:37.512,1:29:40.650
doesn't change the[br]integrand, which is exactly

1:29:40.650,1:29:43.414
the same function, f of xy.

1:29:43.414,1:29:46.810
This is the f function.

1:29:46.810,1:29:47.769
Then what changes?

1:29:47.769,1:29:48.810
The order of integration.

1:29:48.810,1:29:52.168
So I go dx first,[br]dy next and stop.

1:29:52.168,1:29:55.030


1:29:55.030,1:30:00.230
I copy and paste the outer[br]ones, and I focus my attention

1:30:00.230,1:30:05.600
to the red parentheses[br]inside, which I'm

1:30:05.600,1:30:07.890
going to copy and paste here.

1:30:07.890,1:30:12.480
I'll have to do some[br]math very carefully.

1:30:12.480,1:30:13.500
So what do I have?

1:30:13.500,1:30:17.140
I have x plus y integrated[br]with respect to x.

1:30:17.140,1:30:19.232
If I rush, it's a bad thing.

1:30:19.232,1:30:20.800
STUDENT: So that[br]would be x squared.

1:30:20.800,1:30:21.674
PROFESSOR: x squared.

1:30:21.674,1:30:22.780
STUDENT: Over 2.

1:30:22.780,1:30:23.742
PROFESSOR: Over 2.

1:30:23.742,1:30:25.350
STUDENT: Plus xy.

1:30:25.350,1:30:29.610
PROFESSOR: Plus xy taken[br]between the following.

1:30:29.610,1:30:32.760
When x equals 1,[br]I have it on top.

1:30:32.760,1:30:38.350
When x equals y over 2,[br]I have it on the bottom.

1:30:38.350,1:30:39.690
OK.

1:30:39.690,1:30:42.585
This red thing, I'm a[br]little bit too lazy.

1:30:42.585,1:30:47.854
I'll copy and paste[br]it separately.

1:30:47.854,1:30:52.480
For the upper part, it's[br]really easy to compute.

1:30:52.480,1:30:53.480
What do I get?

1:30:53.480,1:31:01.890
When x is 1, 1/2, 1/2[br]plus when x is 1, y.

1:31:01.890,1:31:07.120
Minus integral of--[br]when x is y over 2,

1:31:07.120,1:31:12.700
I get y squared over[br]4 up here over 2.

1:31:12.700,1:31:19.360
So I should get y[br]squared over 8 plus--

1:31:19.360,1:31:21.780
I've got an x equals y over 2.

1:31:21.780,1:31:23.590
What do I get?

1:31:23.590,1:31:26.500
y squared over 2.

1:31:26.500,1:31:28.720
Is this hard?

1:31:28.720,1:31:31.631
It's very easy to make an[br]algebra mistake on such

1:31:31.631,1:31:32.672
a problem, unfortunately.

1:31:32.672,1:31:37.024
I have y plus 1/2 plus what?

1:31:37.024,1:31:40.856
What is 1/2 plus 1/8?

1:31:40.856,1:31:41.820
STUDENT: 5/8.

1:31:41.820,1:31:48.520
PROFESSOR: 5 over 8[br]with a minus y squared.

1:31:48.520,1:31:52.400


1:31:52.400,1:31:54.270
So hopefully I did this right.

1:31:54.270,1:32:00.970
Now I'll go, OK, integral from[br]0 to 2 of all of this animal, y

1:32:00.970,1:32:06.105
plus 1/2 minus 5[br]over 8, y squared.

1:32:06.105,1:32:10.660
What happens if I don't[br]get the right answer?

1:32:10.660,1:32:12.940
Then I go back and[br]check my work because I

1:32:12.940,1:32:14.980
know I'm supposed to get 4/3.

1:32:14.980,1:32:16.150
That was easy.

1:32:16.150,1:32:23.310
So what is integral of this[br]sausage, whatever it is?

1:32:23.310,1:32:29.850
y squared over 2 plus y[br]over 2 minus 5 over 8--

1:32:29.850,1:32:41.930
oh my god-- 5 over 8, y[br]cubed over 3, between 2 up

1:32:41.930,1:32:43.810
and 0 down.

1:32:43.810,1:32:46.730
When I have 0 down,[br]I plug y equals 0.

1:32:46.730,1:32:47.800
It's a piece of cake.

1:32:47.800,1:32:49.146
It's 0.

1:32:49.146,1:32:51.720
So what matters is[br]what I get when I plug

1:32:51.720,1:32:53.560
in the value 2 instead of y.

1:32:53.560,1:32:56.100
So what do I get?

1:32:56.100,1:33:07.020
4 over 2 is 2, plus 2 over 2[br]is 1, minus 2 cubed, thank god.

1:33:07.020,1:33:07.706
That's 8.

1:33:07.706,1:33:10.740
8 simplifies with 8 minus 5/3.

1:33:10.740,1:33:16.280


1:33:16.280,1:33:23.650
So I got 9/3 minus 5/3,[br]and I did it carefully.

1:33:23.650,1:33:25.150
I did a good job.

1:33:25.150,1:33:27.735
I got the same thing, 4/3.

1:33:27.735,1:33:30.880
So no matter which[br]method, the vertical strip

1:33:30.880,1:33:34.390
or the horizontal strip[br]method, I get the same thing.

1:33:34.390,1:33:36.660
And of course, you'll[br]always get the same answer

1:33:36.660,1:33:42.800
because this is what the Fubini[br]theorem extended to this case

1:33:42.800,1:33:43.760
is telling you.

1:33:43.760,1:33:46.762
It doesn't matter the[br]order of integration.

1:33:46.762,1:33:51.230


1:33:51.230,1:33:54.616
I would advise you to go[br]through the theory in the book.

1:33:54.616,1:33:57.950


1:33:57.950,1:34:02.474
They teach you more about[br]area and volume on page 934.

1:34:02.474,1:34:07.910
I'd like you to read that.

1:34:07.910,1:34:11.580
And let's see what I want to do.

1:34:11.580,1:34:14.030
Which one shall I do?

1:34:14.030,1:34:17.950
There are a few examples[br]that are worth it.

1:34:17.950,1:34:21.250


1:34:21.250,1:34:29.010
I'll pick the one that gives[br]people the most trouble.

1:34:29.010,1:34:29.670
How about that?

1:34:29.670,1:34:33.460
I take the few examples that[br]give people the most trouble.

1:34:33.460,1:34:39.130
One example that popped up on[br]almost each and every final

1:34:39.130,1:34:44.490
in the past 13 years[br]that involves changing

1:34:44.490,1:34:46.462
the order of integration.

1:34:46.462,1:34:57.308


1:34:57.308,1:35:10.350
So example problem on changing[br]the order of integration.

1:35:10.350,1:35:14.650


1:35:14.650,1:35:19.615
A very tricky, smart[br]problem is the following.

1:35:19.615,1:35:30.970
Evaluate integral from 0[br]to 1, integral from x to 1,

1:35:30.970,1:35:34.208
e to the y squared dy dx.

1:35:34.208,1:35:41.824


1:35:41.824,1:35:43.740
I don't know if you've[br]seen anything like that

1:35:43.740,1:35:46.135
in AP Calculus or Calc 2.

1:35:46.135,1:35:51.970
Maybe you have, in which case[br]your professor probably told

1:35:51.970,1:35:54.200
you that this is nasty.

1:35:54.200,1:35:57.290


1:35:57.290,1:35:59.515
You say, in what[br]sense is it nasty?

1:35:59.515,1:36:05.050
There is no expressible[br]anti-derivative.

1:36:05.050,1:36:21.600
So this cannot be expressed in[br]terms of elementary functions

1:36:21.600,1:36:22.100
explicitly.

1:36:22.100,1:36:28.587


1:36:28.587,1:36:31.420
It's not that there[br]is no anti-derivative.

1:36:31.420,1:36:34.760
There is an anti-derivative--[br]a whole family, actually--

1:36:34.760,1:36:38.810
but you cannot express them in[br]terms of elementary functions.

1:36:38.810,1:36:42.880
And actually, most functions[br]are not so bad in real world,

1:36:42.880,1:36:44.390
in real life.

1:36:44.390,1:36:48.570
Now, could you compute, for[br]example, integral from 1 to 3

1:36:48.570,1:36:51.290
of e to the t squared dt?

1:36:51.290,1:36:52.100
Yes.

1:36:52.100,1:36:53.524
How do you do that?

1:36:53.524,1:36:55.500
With a calculator.

1:36:55.500,1:36:57.560
And what if you don't have one?

1:36:57.560,1:36:59.060
You go to the lab over there.

1:36:59.060,1:37:00.400
There is MATLAB.

1:37:00.400,1:37:03.105
MATLAB will compute it for you.

1:37:03.105,1:37:04.950
How does MATLAB know[br]how to compute it

1:37:04.950,1:37:07.976
if there is no way to[br]express the anti-derivative

1:37:07.976,1:37:12.035
and take the value of the[br]anti-derivative between b

1:37:12.035,1:37:16.080
and a, like in the fundamental[br]theorem of calculus?

1:37:16.080,1:37:20.705
Well, the calculator or the[br]computer program is smart.

1:37:20.705,1:37:24.700
He uses numerical analysis[br]to approximate this type

1:37:24.700,1:37:26.690
of integral.

1:37:26.690,1:37:27.900
So he's fooling you.

1:37:27.900,1:37:29.850
He's just playing smarty pants.

1:37:29.850,1:37:33.160
He's smarter than[br]you at this point.

1:37:33.160,1:37:33.660
OK.

1:37:33.660,1:37:38.305
So you cannot do this by hand,[br]so this order of integration is

1:37:38.305,1:37:38.805
fruitless.

1:37:38.805,1:37:43.260


1:37:43.260,1:37:47.373
And there are people who[br]tried to do this on the final.

1:37:47.373,1:37:48.831
Of course, they[br]didn't get anywhere

1:37:48.831,1:37:51.062
because they couldn't[br]integrate it.

1:37:51.062,1:37:55.920
The whole idea of this one[br]is to-- some professors

1:37:55.920,1:37:58.690
are so mean they[br]don't even tell you,

1:37:58.690,1:38:00.566
hint, change the[br]order of integration

1:38:00.566,1:38:03.160
because it may work[br]the other way around.

1:38:03.160,1:38:06.260
They just give it to you, and[br]then people can spend an hour

1:38:06.260,1:38:08.420
and they don't get anywhere.

1:38:08.420,1:38:12.145
If you want to be mean to a[br]student, that's what you do.

1:38:12.145,1:38:17.305
So I will tell[br]you that one needs

1:38:17.305,1:38:20.020
to change the order of[br]integration for this.

1:38:20.020,1:38:21.100
This is the function.

1:38:21.100,1:38:26.040
We keep the function, but let's[br]see what happens if you draw.

1:38:26.040,1:38:31.120
The domain will be[br]x between 0 and 1.

1:38:31.120,1:38:33.920
This is your x value.

1:38:33.920,1:38:37.350
y will be between x and 1.

1:38:37.350,1:38:39.990
So it's like you have a square.

1:38:39.990,1:38:43.690
y equals x is your[br]diagonal of the square.

1:38:43.690,1:38:49.420
And you go from--[br]more colors, please.

1:38:49.420,1:38:54.780
You go from y equals x on the[br]bottom and y equals 1 on top.

1:38:54.780,1:38:57.490
And so the domain is[br]this beautiful triangle

1:38:57.490,1:39:02.710
that I make all in line[br]with vertical strips.

1:39:02.710,1:39:06.320
This is what it means,[br]vertical strips.

1:39:06.320,1:39:11.916
But if I do horizontal strips, I[br]have to change the color, blue.

1:39:11.916,1:39:14.545
And for horizontal[br]strips, I'm going

1:39:14.545,1:39:16.696
to have a different problem.

1:39:16.696,1:39:20.220
Integral, integral dx dy.

1:39:20.220,1:39:23.110
And I just hope to god[br]that what I'm going to get

1:39:23.110,1:39:27.000
is doable because if[br]not, then I'm in trouble.

1:39:27.000,1:39:30.280
So help me on this one.

1:39:30.280,1:39:33.680
If y is between what and what?

1:39:33.680,1:39:35.175
It's a square.

1:39:35.175,1:39:38.612
It's a square, so this will[br]be the same, 0 to 1, right?

1:39:38.612,1:39:39.416
STUDENT: Yep.

1:39:39.416,1:39:40.310
PROFESSOR: But Mr. X?

1:39:40.310,1:39:41.715
How about Mr. X?

1:39:41.715,1:39:45.930
STUDENT: And then it[br]will be between 1 and y.

1:39:45.930,1:39:49.410
PROFESSOR: Between--[br]Mr. X is this guy.

1:39:49.410,1:39:51.880
And he doesn't go between 1.

1:39:51.880,1:39:54.844
He goes between the[br]sea level, which is

1:39:54.844,1:40:02.600
x equals 0, to x equals what?

1:40:02.600,1:40:03.490
STUDENT: [INAUDIBLE].

1:40:03.490,1:40:04.772
PROFESSOR: Right?

1:40:04.772,1:40:10.160
So from x equals 0[br]through x equals y.

1:40:10.160,1:40:15.430
And you have the same individual[br]e to the y squared that before

1:40:15.430,1:40:17.220
went on your nerves.

1:40:17.220,1:40:20.130
Now he's not so bad, actually.

1:40:20.130,1:40:21.786
Why is he not so bad?

1:40:21.786,1:40:24.720
Look what happens in[br]the first parentheses.

1:40:24.720,1:40:27.280
This is so beautiful[br]that it's something

1:40:27.280,1:40:29.100
you didn't even hope for.

1:40:29.100,1:40:34.450
So we copy and paste[br]it from 0 to 1 dy.

1:40:34.450,1:40:38.330
These guys stay[br]outside and they wait.

1:40:38.330,1:40:40.370
Inside, it's our[br]business what we do.

1:40:40.370,1:40:44.790
So Mr. X is independent[br]from e to the y squared.

1:40:44.790,1:40:46.830
So e to the y squared pulls out.

1:40:46.830,1:40:48.342
He's a constant.

1:40:48.342,1:40:53.390
And you have integral[br]of 1 dx between 0 and y.

1:40:53.390,1:40:56.670
How much is that?

1:40:56.670,1:40:57.920
1.

1:40:57.920,1:41:03.550
x between x equals[br]0 and x equals y.

1:41:03.550,1:41:05.001
So it's y.

1:41:05.001,1:41:05.875
So I'm being serious.

1:41:05.875,1:41:07.600
So I should have said y.

1:41:07.600,1:41:11.830


1:41:11.830,1:41:21.050
Now, if your professor would[br]have given you, in Calc 2,

1:41:21.050,1:41:25.050
this, how would[br]you have done it?

1:41:25.050,1:41:26.927
STUDENT: U-substitution.

1:41:26.927,1:41:28.010
PROFESSOR: U-substitution.

1:41:28.010,1:41:28.776
Excellent.

1:41:28.776,1:41:32.130
What kind of[br]u-substitution [INAUDIBLE]?

1:41:32.130,1:41:35.100
STUDENT: y squared equals u.

1:41:35.100,1:41:40.410
PROFESSOR: y squared[br]equals u, du equals 2y dy.

1:41:40.410,1:41:44.400
So y dy together.

1:41:44.400,1:41:45.973
They stick together.

1:41:45.973,1:41:47.332
They stick together.

1:41:47.332,1:41:49.560
They attract each[br]other as magnets.

1:41:49.560,1:41:56.932
So y dy is going to be[br]1/2 du-- 1/2 pulls out--

1:41:56.932,1:42:00.190
integral e to the u du.

1:42:00.190,1:42:00.690
Attention.

1:42:00.690,1:42:03.430
When y is moving[br]between 0 and 1,

1:42:03.430,1:42:06.211
u is moving also[br]between 0 and 1.

1:42:06.211,1:42:11.990
So it really should[br]be a piece of cake.

1:42:11.990,1:42:13.495
Are you guys with me?

1:42:13.495,1:42:15.880
Do you understand what I did?

1:42:15.880,1:42:18.970
Do you understand the words[br]coming out of my mouth?

1:42:18.970,1:42:24.415


1:42:24.415,1:42:25.405
It's easy.

1:42:25.405,1:42:29.490


1:42:29.490,1:42:29.990
Good.

1:42:29.990,1:42:35.460
So what is integral[br]of e to the u du?

1:42:35.460,1:42:39.220
e to the u between[br]1 up and 0 down.

1:42:39.220,1:42:43.983
So e to the u de to the 1[br]minus e to the 0 over 2.

1:42:43.983,1:42:47.930


1:42:47.930,1:42:51.045
That is e minus 1 over 2.

1:42:51.045,1:42:54.320


1:42:54.320,1:42:59.215
I could not have solved this[br]if I tried it by integration

1:42:59.215,1:43:02.670
with y first and then x.

1:43:02.670,1:43:04.600
The only way I[br]could have done this

1:43:04.600,1:43:07.700
is by changing the[br]order of integration.

1:43:07.700,1:43:11.556
So how many times have I seen[br]this in the past 12 years

1:43:11.556,1:43:12.510
on the final?

1:43:12.510,1:43:15.228
At least six times.

1:43:15.228,1:43:18.220
It's a problem that[br]could be a little bit

1:43:18.220,1:43:21.150
hard if the student has[br]never seen it before

1:43:21.150,1:43:23.810
and doesn't know what to[br]do [? at that point. ?]

1:43:23.810,1:43:27.075
Let's do a few more[br]in the same category.

1:43:27.075,1:43:36.648


1:43:36.648,1:43:37.612
STUDENT: Professor?

1:43:37.612,1:43:38.266
PROFESSOR: Yes?

1:43:38.266,1:43:40.890
STUDENT: Where did this shape--[br]where did this graph come from?

1:43:40.890,1:43:43.720
Were we just saying[br]it was with the same--

1:43:43.720,1:43:44.640
PROFESSOR: OK.

1:43:44.640,1:43:46.750
I read it from here.

1:43:46.750,1:43:50.460
So this and that are the key.

1:43:50.460,1:43:55.160
This is telling me x is between[br]0 and 1, and at the same,

1:43:55.160,1:43:59.010
time y is between x and 1.

1:43:59.010,1:44:02.510
And when I read this[br]information on the graph,

1:44:02.510,1:44:05.570
I say, well, x is[br]between 0 and 1.

1:44:05.570,1:44:09.070
Mr. Y has the freedom to go[br]between the first bisector,

1:44:09.070,1:44:14.070
which is that, and the[br]cap, his cap, y equals 1.

1:44:14.070,1:44:17.450
So that's how I got[br]to the line strips.

1:44:17.450,1:44:21.265
And from the line strips, I said[br]that I need horizontal strips.

1:44:21.265,1:44:23.970
So I changed the[br]color and I said

1:44:23.970,1:44:27.692
the blue strips go between x.

1:44:27.692,1:44:31.856
x will be x equals[br]0 and x equals y.

1:44:31.856,1:44:36.910
And then y between 0[br]and 1, just the same.

1:44:36.910,1:44:38.080
It's a little bit tricky.

1:44:38.080,1:44:42.360
That's why I want to do one or[br]two more problems like that,

1:44:42.360,1:44:46.840
because I know that I remember[br]20-something years ago,

1:44:46.840,1:44:52.500
I myself needed a little[br]bit of time understanding

1:44:52.500,1:44:56.710
the meaning of reversing[br]the order of integration.

1:44:56.710,1:44:58.970
STUDENT: Does it matter[br]which way you put it?

1:44:58.970,1:45:02.410
PROFESSOR: In this case, it's[br]important that you do reverse.

1:45:02.410,1:45:05.890
But in general, it's[br]doable both ways.

1:45:05.890,1:45:10.018
I mean, in the other problems[br]I'm going to give you today,

1:45:10.018,1:45:11.890
you should be able[br]to do either way.

1:45:11.890,1:45:19.134
So I'm looking for a problem[br]that you could eventually

1:45:19.134,1:45:20.610
do another one.

1:45:20.610,1:45:25.530


1:45:25.530,1:45:27.990
We don't have so many.

1:45:27.990,1:45:31.716
I'm going to go ahead and[br]look into the homework.

1:45:31.716,1:45:32.215
Yeah.

1:45:32.215,1:45:34.835


1:45:34.835,1:45:41.395
So it says, you[br]have this integral,

1:45:41.395,1:45:44.170
the integral from 0[br]to 4 of the integral

1:45:44.170,1:45:49.880
from x squared to 4y dy dx.

1:45:49.880,1:45:55.990
Draw, compute, and also[br]compute with reversing

1:45:55.990,1:45:58.990
the order of integration[br]to check your work.

1:45:58.990,1:46:01.160
When I say that,[br]it sounds horrible.

1:46:01.160,1:46:04.280
But in reality, the[br]more you work on

1:46:04.280,1:46:08.116
that one, the more familiar[br]you're going to feel.

1:46:08.116,1:46:10.500
So what did I just say?

1:46:10.500,1:46:12.740
Problem number 26.

1:46:12.740,1:46:18.200
You have integral[br]from 0 to 4, integral

1:46:18.200,1:46:24.560
from x squared to 4x dy dx.

1:46:24.560,1:46:27.400


1:46:27.400,1:46:31.310
Interpret geometrically,[br]whatever that means,

1:46:31.310,1:46:35.255
and then compute the[br]integral in two ways,

1:46:35.255,1:46:37.867
with this given order[br]integration, which

1:46:37.867,1:46:40.005
is what kind of strips, guys?

1:46:40.005,1:46:41.910
Vertical strips.

1:46:41.910,1:46:45.010
Or reversing the[br]order of integration.

1:46:45.010,1:46:50.222
And check that the answer is the[br]same just to check your work.

1:46:50.222,1:46:51.960
STUDENT: So first--

1:46:51.960,1:46:53.100
PROFESSOR: First you draw.

1:46:53.100,1:46:55.890
First you draw because[br]if you don't draw,

1:46:55.890,1:47:00.490
you don't understand what[br]the problem is about.

1:47:00.490,1:47:01.730
And you say, wait a minute.

1:47:01.730,1:47:05.290
But couldn't I go ahead[br]and do it without drawing?

1:47:05.290,1:47:08.400
Yeah, but you're not[br]going to get too far.

1:47:08.400,1:47:11.870
So let's see what kind[br]of problem you have.

1:47:11.870,1:47:13.210
y and x.

1:47:13.210,1:47:16.800
y equals x squared is a what?

1:47:16.800,1:47:18.758
It's a pa--

1:47:18.758,1:47:19.746
STUDENT: Parabola.

1:47:19.746,1:47:20.734
PROFESSOR: Parabola.

1:47:20.734,1:47:24.192
And this parabola should[br]be nice and sassy.

1:47:24.192,1:47:25.680
Is it fat enough?

1:47:25.680,1:47:27.470
I think it is.

1:47:27.470,1:47:34.140
And the other one will[br]be 4x, y equals 4x.

1:47:34.140,1:47:36.070
What does that look like?

1:47:36.070,1:47:39.340
It looks like a line passing[br]through the origin that

1:47:39.340,1:47:42.700
has slope 4, so the[br]slope is really high.

1:47:42.700,1:47:43.690
STUDENT: Just straight.

1:47:43.690,1:47:48.150


1:47:48.150,1:47:51.870
PROFESSOR: y equals 4x[br]versus y equals x squared.

1:47:51.870,1:47:53.690
Now, do they meet?

1:47:53.690,1:47:57.343


1:47:57.343,1:47:57.884
STUDENT: Yes.

1:47:57.884,1:47:58.508
PROFESSOR: Yes.

1:47:58.508,1:47:59.750
Exactly where do they meet?

1:47:59.750,1:48:00.300
Exactly here.

1:48:00.300,1:48:00.800
STUDENT: 4.

1:48:00.800,1:48:04.190
PROFESSOR: So 4x equals x[br]squared, where do they meet?

1:48:04.190,1:48:06.930


1:48:06.930,1:48:13.480
They meet at-- it has[br]two possible roots.

1:48:13.480,1:48:18.420
One is x equals[br]0, which is here,

1:48:18.420,1:48:21.270
and one is x equals[br]4, which is here.

1:48:21.270,1:48:26.720
So really, my graph looks[br]just the way it should look,

1:48:26.720,1:48:29.450
only my parabola is[br]a little bit too fat.

1:48:29.450,1:48:33.840


1:48:33.840,1:48:44.100
This is the point of[br]coordinates 4 and 16.

1:48:44.100,1:48:46.410
Are you guys with me?

1:48:46.410,1:48:52.350
And Mr. X is moving[br]between 0 and 4.

1:48:52.350,1:48:56.910
This is the maximum[br]level x can get.

1:48:56.910,1:49:01.730
And where he stops here[br]at 4, a miracle happens.

1:49:01.730,1:49:06.680
The two curves intersect each[br]other exactly at that point.

1:49:06.680,1:49:11.904
So this looks like a[br]leaf, a slice of orange.

1:49:11.904,1:49:12.404
Oh my god.

1:49:12.404,1:49:12.945
I don't know.

1:49:12.945,1:49:17.825
I'm already hungry so I cannot[br]wait to get out of here.

1:49:17.825,1:49:20.675
I bet you're hungry as well.

1:49:20.675,1:49:24.000
Let's do this problem[br]both ways and then go

1:49:24.000,1:49:26.537
home or to have[br]something to eat.

1:49:26.537,1:49:31.958
How are you going to advise[br]me to solve it first?

1:49:31.958,1:49:34.220
It's already set[br]up to be solved.

1:49:34.220,1:49:35.476
So it's vertical strips.

1:49:35.476,1:49:37.910
And I will say[br]integral from 0 to 4,

1:49:37.910,1:49:40.790
copy and paste the outer part.

1:49:40.790,1:49:46.090
Take the inner part, and do the[br]inner part because it's easy.

1:49:46.090,1:49:50.402
And if it's easy, you tell[br]me how I'm going to do it.

1:49:50.402,1:49:53.556
Integral of 1 dy is y.

1:49:53.556,1:49:58.820
y measured at 4x is 4x,[br]and y measured at x squared

1:49:58.820,1:50:01.120
is x squared.

1:50:01.120,1:50:01.790
Oh thank god.

1:50:01.790,1:50:05.728
This is so beautiful[br]and so easy.

1:50:05.728,1:50:08.650
Let's integrate again.

1:50:08.650,1:50:16.370
4 x squared over 2 times x cubed[br]over 3 between x equals 0 down

1:50:16.370,1:50:17.920
and x equals 4 up.

1:50:17.920,1:50:22.190


1:50:22.190,1:50:23.810
What do I get?

1:50:23.810,1:50:30.050
I get 4 cubed over 2[br]minus 4 cubed over 3.

1:50:30.050,1:50:31.840
This 4 cubed is an obsession.

1:50:31.840,1:50:33.814
Kick him out.

1:50:33.814,1:50:35.742
1/2 minus 1/3.

1:50:35.742,1:50:39.610


1:50:39.610,1:50:41.450
How much is 1/2 minus 1/3?

1:50:41.450,1:50:42.430
My son knows that.

1:50:42.430,1:50:43.577
STUDENT: 1/6.

1:50:43.577,1:50:44.160
PROFESSOR: OK.

1:50:44.160,1:50:46.060
1/6, yes.

1:50:46.060,1:50:48.710
So we simply take it.

1:50:48.710,1:50:49.965
We can leave it like that.

1:50:49.965,1:50:55.530
If you leave it like that on[br]the exam, I don't mind at all.

1:50:55.530,1:50:58.681
But you could always put[br]64 over 6 and simplify it.

1:50:58.681,1:51:01.507


1:51:01.507,1:51:03.391
Are you guys with me?

1:51:03.391,1:51:07.070
You can simplify[br]it and get what?

1:51:07.070,1:51:08.202
32 over 3.

1:51:08.202,1:51:10.980


1:51:10.980,1:51:12.530
Don't give me decimals.

1:51:12.530,1:51:14.634
I'm not impressed.

1:51:14.634,1:51:16.383
You're not supposed[br]to use the calculator.

1:51:16.383,1:51:21.295
You are supposed to leave[br]this is exact fraction

1:51:21.295,1:51:24.850
form like that, irreducible.

1:51:24.850,1:51:26.320
Let's do it the[br]other way around,

1:51:26.320,1:51:30.020
and that will be the[br]last thing we do.

1:51:30.020,1:51:34.020
The other way around means[br]I'll take another color.

1:51:34.020,1:51:36.980
I'll do the horizontal stripes.

1:51:36.980,1:51:40.010


1:51:40.010,1:51:44.110
And I will have to rewrite[br]the meaning of these two

1:51:44.110,1:51:49.530
branches of functions with[br]x expressed in terms of y.

1:51:49.530,1:51:51.710
That's the only thing[br]I need to do, right?

1:51:51.710,1:51:55.910
So what is this?

1:51:55.910,1:51:59.210
If y is x squared, what is x?

1:51:59.210,1:52:00.150
STUDENT: Root y.

1:52:00.150,1:52:03.735
PROFESSOR: The inverse[br]function. x will be root of y.

1:52:03.735,1:52:06.000
You said very well.

1:52:06.000,1:52:07.360
So I have to write.

1:52:07.360,1:52:10.470
In [INAUDIBLE], I[br]have what I need

1:52:10.470,1:52:13.082
to have for the line[br]horizontal strip method.

1:52:13.082,1:52:16.060


1:52:16.060,1:52:19.655
And then for the other one,[br]x is going to be y over 4.

1:52:19.655,1:52:22.610


1:52:22.610,1:52:23.600
So what do I do?

1:52:23.600,1:52:32.382
So integral, integral, a[br]1 that was here hidden,

1:52:32.382,1:52:35.826
but I'll put it because[br]that's the integral.

1:52:35.826,1:52:38.560
And then I go dx dy.

1:52:38.560,1:52:44.990
All I have to care about is the[br]endpoints of the integration.

1:52:44.990,1:52:48.240
Now, pay attention a little[br]bit because Mr. Y is not

1:52:48.240,1:52:49.660
between 0 and 4.

1:52:49.660,1:52:53.210
I had very good[br]students under stress

1:52:53.210,1:52:55.620
in the final putting 0 and 4.

1:52:55.620,1:52:56.760
Don't do that.

1:52:56.760,1:52:59.245
So pay attention to the[br]limits of integration.

1:52:59.245,1:53:01.030
What are the limits?

1:53:01.030,1:53:01.750
0 and--

1:53:01.750,1:53:02.407
STUDENT: 16.

1:53:02.407,1:53:02.990
PROFESSOR: 16.

1:53:02.990,1:53:04.970
Very good.

1:53:04.970,1:53:09.610
And x will be between root[br]y-- well, which one is on top?

1:53:09.610,1:53:11.610
Which one is on the bottom?

1:53:11.610,1:53:17.110
Because if I move my head,[br]I'll say that's on top

1:53:17.110,1:53:18.610
and that's on the bottom.

1:53:18.610,1:53:22.210
STUDENT: The right side[br]is always on the top.

1:53:22.210,1:53:25.730
PROFESSOR: So the one that[br]looks higher is this one.

1:53:25.730,1:53:29.210
This is more than[br]that in this frame.

1:53:29.210,1:53:36.583
So square of y is on top and[br]y over 4 is on the bottom.

1:53:36.583,1:53:38.978
I should get the same answer.

1:53:38.978,1:53:40.420
If I don't, then I'm in trouble.

1:53:40.420,1:53:43.216
So what do I get?

1:53:43.216,1:53:49.084
Integral from 0 to 16.

1:53:49.084,1:53:51.500
Tonight, when I[br]go home, I'm going

1:53:51.500,1:53:57.024
to cook up the homework[br]for 12.1 and 12.1 at least.

1:53:57.024,1:53:59.175
I'll put some problems[br]similar to that

1:53:59.175,1:54:02.590
because I want to emphasize[br]the same type of problem

1:54:02.590,1:54:05.010
in at least two or three[br]applications for the homework

1:54:05.010,1:54:07.200
for the midterm.

1:54:07.200,1:54:10.575
And maybe one like that will[br]be on the final as well.

1:54:10.575,1:54:13.285
It's very important for[br]you to understand how,

1:54:13.285,1:54:15.273
with this kind of[br]domain, you reverse

1:54:15.273,1:54:16.834
the order of integration.

1:54:16.834,1:54:19.760
Who's helping me here?

1:54:19.760,1:54:22.210
Root y.

1:54:22.210,1:54:26.070
What is root y[br]when-- y to the 1/2.

1:54:26.070,1:54:28.070
I need to integrate.

1:54:28.070,1:54:33.552
So I need minus y over 4 and dy.

1:54:33.552,1:54:39.030


1:54:39.030,1:54:42.188
Can you help me integrate?

1:54:42.188,1:54:44.180
STUDENT: [INAUDIBLE].

1:54:44.180,1:54:49.830
PROFESSOR: 2/3 y[br]to the 3/2 minus--

1:54:49.830,1:54:51.120
STUDENT: y squared.

1:54:51.120,1:54:56.480
PROFESSOR: y squared[br]over 8, y equals 0

1:54:56.480,1:54:58.490
on the bottom, piece of cake.

1:54:58.490,1:55:00.220
That will give me 0.

1:55:00.220,1:55:00.966
I'm so happy.

1:55:00.966,1:55:04.560
And y equals 16 on top.

1:55:04.560,1:55:09.830
So for 16, I have 2/3.

1:55:09.830,1:55:12.210
And who's telling me what else?

1:55:12.210,1:55:13.170
STUDENT: 64.

1:55:13.170,1:55:13.910
PROFESSOR: 64.

1:55:13.910,1:55:14.480
4 cubed.

1:55:14.480,1:55:22.700
I can leave it 4 cubed if I want[br]to minus another-- well here,

1:55:22.700,1:55:24.740
I have to pay attention.

1:55:24.740,1:55:27.350
So I have 16 here.

1:55:27.350,1:55:31.390
I got square root of[br]16, which is 4, cubed.

1:55:31.390,1:55:38.720
Here, I put minus 4[br]squared, which was there.

1:55:38.720,1:55:40.635
How do you want me to[br]do this simplification?

1:55:40.635,1:55:41.910
STUDENT: [INAUDIBLE].

1:55:41.910,1:55:44.840
PROFESSOR: I can[br]do 4 to the fourth.

1:55:44.840,1:55:47.190
Are you guys with me?

1:55:47.190,1:55:52.400
I can put, like you[br]prefer, 16 squared over 8.

1:55:52.400,1:55:57.900


1:55:57.900,1:55:59.390
Is it the same answer?

1:55:59.390,1:56:00.150
I don't know.

1:56:00.150,1:56:02.410
Let's see.

1:56:02.410,1:56:09.050
This is really 4 to the 4,[br]so I have 4 times 4 cubed.

1:56:09.050,1:56:19.788
4 cubed gets out and[br]I have 2/3 minus 1/2.

1:56:19.788,1:56:24.060


1:56:24.060,1:56:27.640
And how much is that?

1:56:27.640,1:56:28.765
Again 1/6.

1:56:28.765,1:56:30.670
Are you guys with me?

1:56:30.670,1:56:31.620
1/6.

1:56:31.620,1:56:36.860
So again, I get 4 cubed[br]over 6, so I'm done.

1:56:36.860,1:56:40.420
4 cubed over 6 equals 32 over 3.

1:56:40.420,1:56:42.960
I am happy that[br]I checked my work

1:56:42.960,1:56:44.420
through two different methods.

1:56:44.420,1:56:45.710
I got the same answer.

1:56:45.710,1:56:49.220


1:56:49.220,1:56:51.500
Now, let me tell you something.

1:56:51.500,1:56:55.220
There were also times[br]when on the midterm

1:56:55.220,1:56:59.800
or on the final, due to[br]lack of time and everything,

1:56:59.800,1:57:02.930
we put the following[br]kind of problem.

1:57:02.930,1:57:11.290
Without solving this integral--[br]without solving-- indicate

1:57:11.290,1:57:16.290
the corresponding integral[br]with the order reversed.

1:57:16.290,1:57:19.680
So all you have to[br]do-- don't do that.

1:57:19.680,1:57:24.715
Just from here,[br]write this and stop.

1:57:24.715,1:57:27.579
Don't waste your time.

1:57:27.579,1:57:29.620
If you do the whole thing,[br]it's going to take you

1:57:29.620,1:57:30.585
10 minutes, 15 minutes.

1:57:30.585,1:57:33.902
If you do just reversing[br]the order of integration,

1:57:33.902,1:57:38.265
I don't know what it takes, a[br]minute and a half, two minutes.

1:57:38.265,1:57:42.250
So in order to save[br]time, at times,

1:57:42.250,1:57:46.166
we gave you just don't[br]solve the problem. reverse

1:57:46.166,1:57:47.618
the order of integration.

1:57:47.618,1:57:54.400


1:57:54.400,1:57:55.750
One last one.

1:57:55.750,1:57:58.230
One last one.

1:57:58.230,1:57:59.730
But I don't want to finish it.

1:57:59.730,1:58:03.226
I want to give you[br]the answer at home,

1:58:03.226,1:58:05.681
or maybe you can finish it.

1:58:05.681,1:58:07.645
It should be shorter.

1:58:07.645,1:58:13.537
You have a circular parabola,[br]but only the first quadrant.

1:58:13.537,1:58:16.490


1:58:16.490,1:58:19.253
So x is positive.

1:58:19.253,1:58:20.260
STUDENT: Question.

1:58:20.260,1:58:21.260
PROFESSOR: I don't know.

1:58:21.260,1:58:22.480
I have to find it.

1:58:22.480,1:58:23.840
Find the volume.

1:58:23.840,1:58:25.470
Example 4, page 934.

1:58:25.470,1:58:28.560
Find the volume[br]of the solid bound

1:58:28.560,1:58:32.690
in the above-- this is a[br]little tricky-- by the plane z

1:58:32.690,1:58:38.010
equals y and below[br]in the xy plane

1:58:38.010,1:58:42.460
by the part of the disk[br]in the first quadrant.

1:58:42.460,1:58:47.940
So z equals y means this[br]is your f of x and y.

1:58:47.940,1:58:50.550
So they gave it to you.

1:58:50.550,1:58:54.430
But then they say, but[br]also, in the xy plane,

1:58:54.430,1:59:00.120
you have to have the part of[br]the disk in the first quadrant.

1:59:00.120,1:59:01.790
This is not so easy.

1:59:01.790,1:59:04.637
They draw it for you to[br]make your life easier.

1:59:04.637,1:59:08.050
The first quadrant is that.

1:59:08.050,1:59:13.600
How do you write the unit[br]circle, x squared equals 1,

1:59:13.600,1:59:16.720
x squared plus y squared[br]less than or equal to 1,

1:59:16.720,1:59:19.350
and x and y are both positive.

1:59:19.350,1:59:21.123
This is the first quadrant.

1:59:21.123,1:59:22.512
How do you compute?

1:59:22.512,1:59:26.680
So they say compute the[br]volume, and I say just

1:59:26.680,1:59:27.850
set up the volume.

1:59:27.850,1:59:30.064
Forget about computing it.

1:59:30.064,1:59:33.473
I could put it in the[br]midterm just like that.

1:59:33.473,1:59:36.380
Set up an integral[br]without solving it

1:59:36.380,1:59:46.110
that indicates the volume[br]under z equals f of xy-- that's

1:59:46.110,1:59:50.980
the geography of z-- and above[br]a certain domain in plane,

1:59:50.980,1:59:55.510
above D in plane.

1:59:55.510,1:59:58.090
So you have, OK, what[br]this should teach you?

1:59:58.090,2:00:08.660
Should teach you that double[br]integral over d f of xy da

2:00:08.660,2:00:10.990
can be solved.

2:00:10.990,2:00:12.550
Do I ask to be solved?

2:00:12.550,2:00:13.720
No.

2:00:13.720,2:00:14.445
Why?

2:00:14.445,2:00:18.100
Because you can finish[br]it later, finish at home.

2:00:18.100,2:00:27.230
Or maybe, I don't even want[br]you to compute on the final.

2:00:27.230,2:00:29.280
So how do we do that?

2:00:29.280,2:00:32.970
f is y.

2:00:32.970,2:00:36.520
Would I be able to choose[br]whichever order integration I

2:00:36.520,2:00:38.380
want?

2:00:38.380,2:00:40.000
It shouldn't matter which order.

2:00:40.000,2:00:43.020
It should be more[br]or less the same.

2:00:43.020,2:00:44.695
What if I do dy dx?

2:00:44.695,2:00:47.630


2:00:47.630,2:00:52.248
Then I have to do the Fubini.

2:00:52.248,2:00:54.230
But it's not a[br]rectangular domain.

2:00:54.230,2:00:54.730
Aha.

2:00:54.730,2:00:56.630
So Magdalena, be a[br]little bit careful

2:00:56.630,2:01:00.410
because this is going to[br]be two finite numbers,

2:01:00.410,2:01:01.865
but these are functions.

2:01:01.865,2:01:04.340
STUDENT: It will[br]be an x function.

2:01:04.340,2:01:08.068
PROFESSOR: So the x[br]is between 0 and 1,

2:01:08.068,2:01:10.020
and that's going to be z.

2:01:10.020,2:01:11.484
You do vertical strips.

2:01:11.484,2:01:13.924
That's a piece of cake.

2:01:13.924,2:01:17.890
But if you do the[br]vertical strips,

2:01:17.890,2:01:21.980
you have to pay attention to[br]the endpoints for x and y,

2:01:21.980,2:01:23.426
and one is easy.

2:01:23.426,2:01:24.487
Which one is trivial?

2:01:24.487,2:01:25.070
STUDENT: Zero.

2:01:25.070,2:01:26.403
PROFESSOR: The bottom one, zero.

2:01:26.403,2:01:29.350
The one that's nontrivial[br]is the upper one.

2:01:29.350,2:01:31.290
STUDENT: There will be 1 minus--

2:01:31.290,2:01:33.595
STUDENT: Square root[br]of 1 minus y squared.

2:01:33.595,2:01:34.470
PROFESSOR: Very good.

2:01:34.470,2:01:36.210
Square root of 1[br]minus y squared.

2:01:36.210,2:01:41.110


2:01:41.110,2:01:46.640
So if I were to go one more step[br]further without solving this,

2:01:46.640,2:01:51.400
I'm going to ask you, could[br]this be solved by hand?

2:01:51.400,2:01:57.890
Well, so you have[br]it in the book--

2:01:57.890,2:02:00.390
STUDENT: Professor, should be[br]a [INAUDIBLE] minus x squared?

2:02:00.390,2:02:03.009


2:02:03.009,2:02:03.800
PROFESSOR: Oh yeah.

2:02:03.800,2:02:04.860
1 minus x squared.

2:02:04.860,2:02:06.590
Excuse me.

2:02:06.590,2:02:08.190
Didn't I write it?

2:02:08.190,2:02:11.996
Yeah, here I should have written[br]y equals square root of 1

2:02:11.996,2:02:14.300
minus x squared.

2:02:14.300,2:02:21.263
So when you do it-- thank you[br]so much-- you go integrate,

2:02:21.263,2:02:26.790
and you have y squared over 2.

2:02:26.790,2:02:29.605
And you evaluate[br]between y equals 0

2:02:29.605,2:02:33.466
and y equals square[br]root 1 minus x squared,

2:02:33.466,2:02:34.918
and then you do the [INAUDIBLE].

2:02:34.918,2:02:41.220


2:02:41.220,2:02:44.764
In the book, they[br]do it differently.

2:02:44.764,2:02:50.328
They do it with respect to[br]dx and dy and integrate.

2:02:50.328,2:02:52.798
But it doesn't[br]matter how you do it.

2:02:52.798,2:02:54.774
You should get the same answer.

2:02:54.774,2:02:58.330


2:02:58.330,2:03:00.135
All right?

2:03:00.135,2:03:01.065
[INAUDIBLE]?

2:03:01.065,2:03:03.855
STUDENT: [INAUDIBLE][br]in that way,

2:03:03.855,2:03:06.820
doesn't the square root work out[br]better because there's already

2:03:06.820,2:03:07.690
a y there?

2:03:07.690,2:03:09.036
PROFESSOR: In the other case--

2:03:09.036,2:03:10.900
STUDENT: Doing dy dx.

2:03:10.900,2:03:12.720
PROFESSOR: Yeah,[br]in the other way,

2:03:12.720,2:03:14.340
it works a little[br]bit differently.

2:03:14.340,2:03:17.420
You can do[br]u-substitution, I think.

2:03:17.420,2:03:20.460
So if you do it the other[br]way, it will be what?

2:03:20.460,2:03:24.430
Integral from 0 to[br]1, integral form 0

2:03:24.430,2:03:32.432
to square root of 1[br]minus y squared, y dx dy.

2:03:32.432,2:03:35.166
And what do you do in this case?

2:03:35.166,2:03:37.430
You have integral from 0 to 1.

2:03:37.430,2:03:42.770
Integral of y dx is going[br]to be y is a constant.

2:03:42.770,2:03:48.860
x between the two values will[br]be simply 1 minus y squared dy.

2:03:48.860,2:03:49.920
So you're right.

2:03:49.920,2:03:52.580
Matthew saw that,[br]because he's a prophet,

2:03:52.580,2:03:56.090
and he could see[br]two steps ahead.

2:03:56.090,2:03:57.850
This is very nice[br]what you observed.

2:03:57.850,2:03:59.100
What do you do?

2:03:59.100,2:04:02.590
You take a u-substitution[br]when you go home.

2:04:02.590,2:04:06.010
You get u equals[br]1 minus y squared.

2:04:06.010,2:04:12.798
du will be minus 2y[br]dy, and you go on.

2:04:12.798,2:04:17.172
So in the book, we got 1/3.

2:04:17.172,2:04:19.602
If you continue[br]with this method,

2:04:19.602,2:04:20.907
I think it's the same answer.

2:04:20.907,2:04:21.490
STUDENT: Yeah.

2:04:21.490,2:04:21.989
I got 1/3.

2:04:21.989,2:04:23.060
PROFESSOR: You got 1/3.

2:04:23.060,2:04:26.070
So sounds good.

2:04:26.070,2:04:28.140
We will stop here.

2:04:28.140,2:04:29.730
You will get homework.

2:04:29.730,2:04:32.560
How long should I[br]leave that homework on?

2:04:32.560,2:04:35.810
Because I'm thinking maybe[br]another month, but please

2:04:35.810,2:04:38.190
don't procrastinate.

2:04:38.190,2:04:41.430
So let's say until[br]the end of March.

2:04:41.430,2:04:44.480
And keep in mind[br]that we have included

2:04:44.480,2:04:48.080
one week of spring[br]break here, which you

2:04:48.080,2:04:51.360
can do whatever you want with.

2:04:51.360,2:04:57.740
Some of you may be in Florida[br]swimming and working on a tan,

2:04:57.740,2:04:59.240
and not working on homework.

2:04:59.240,2:05:01.940
So no matter how, plan ahead.

2:05:01.940,2:05:03.440
Plan ahead and you will do well.

2:05:03.440,2:05:10.390
31st of March for[br]the whole chapter.

2:05:10.390,2:05:11.399