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PROFESSOR: Today we're going
to continue with integration.
00:00:24.280 --> 00:00:29.670
And we get to do the-- probably
the most important thing
00:00:29.670 --> 00:00:31.090
of this entire course.
00:00:31.090 --> 00:00:33.970
Which is appropriately named.
00:00:33.970 --> 00:00:50.250
It's called the fundamental
theorem of calculus.
00:00:50.250 --> 00:00:54.900
And we'll be abbreviating
it FTC and occasionally I'll
00:00:54.900 --> 00:00:58.964
put in a 1 here, because there
will be two versions of it.
00:00:58.964 --> 00:01:00.380
But this is the
one that you'll be
00:01:00.380 --> 00:01:06.420
using the most in this class.
00:01:06.420 --> 00:01:14.540
The fundamental theorem of
calculus says the following.
00:01:14.540 --> 00:01:26.875
It says that if F' =
f, so F'(x) = f(x),
00:01:26.875 --> 00:01:30.890
there's a capital
F and a little f,
00:01:30.890 --> 00:01:45.960
then the integral from a to b
of f(x) is equal to F(b) - F(a).
00:01:52.160 --> 00:01:52.940
That's it.
00:01:52.940 --> 00:01:55.600
That's the whole theorem.
00:01:55.600 --> 00:02:00.120
And you may recognize it.
00:02:00.120 --> 00:02:03.880
Before, we had the
notation that F
00:02:03.880 --> 00:02:10.020
was the antiderivative,
that is, capital F
00:02:10.020 --> 00:02:11.635
was the integral of f(x).
00:02:11.635 --> 00:02:12.510
We wrote it this way.
00:02:12.510 --> 00:02:14.860
This is this
indefinite integral.
00:02:14.860 --> 00:02:17.970
And now we're putting
in definite values.
00:02:17.970 --> 00:02:20.260
And we have a connection
between the two
00:02:20.260 --> 00:02:22.420
uses of the integral sign.
00:02:22.420 --> 00:02:24.530
But with the definite
values, we get real numbers
00:02:24.530 --> 00:02:26.220
out instead of a function.
00:02:26.220 --> 00:02:29.810
Or a function up to a constant.
00:02:29.810 --> 00:02:30.990
So this is it.
00:02:30.990 --> 00:02:32.120
This is the formula.
00:02:32.120 --> 00:02:35.400
And it's usually also written
with another notation.
00:02:35.400 --> 00:02:40.390
So I want to introduce that
notation to you as well.
00:02:40.390 --> 00:02:44.950
So there's a new notation here.
00:02:44.950 --> 00:02:47.320
Which you'll find
very convenient.
00:02:47.320 --> 00:02:51.010
Because we don't always
have to give a letter f
00:02:51.010 --> 00:02:52.710
to the functions involved.
00:02:52.710 --> 00:02:54.930
So it's an abbreviation.
00:02:54.930 --> 00:02:57.610
For right now there'll be
a lot of f's, but anyway.
00:02:57.610 --> 00:02:59.660
So here's the abbreviation.
00:02:59.660 --> 00:03:04.420
Whenever I have a difference
between a function at two
00:03:04.420 --> 00:03:09.430
values, I also can
write this as F(x)
00:03:09.430 --> 00:03:12.440
with an a down here
and a b up there.
00:03:12.440 --> 00:03:16.550
So that's the
notation that we use.
00:03:16.550 --> 00:03:19.910
And you can also, for
emphasis, and this sometimes
00:03:19.910 --> 00:03:23.450
turns out to be important, when
there's more than one variable
00:03:23.450 --> 00:03:25.750
floating around in the problem.
00:03:25.750 --> 00:03:28.200
To specify that
the variable is x.
00:03:28.200 --> 00:03:32.270
So this is the same
thing as x = a.
00:03:32.270 --> 00:03:34.100
And x = b.
00:03:34.100 --> 00:03:36.040
It indicates where
you want to plug in,
00:03:36.040 --> 00:03:37.450
what you want to plug in.
00:03:37.450 --> 00:03:41.840
And now you take the top
value minus the bottom value.
00:03:41.840 --> 00:03:43.380
So F(b) - F(a).
00:03:43.380 --> 00:03:50.290
So this is just a notation, and
in that notation, of course,
00:03:50.290 --> 00:03:59.420
the theorem can be written
with this set of symbols here.
00:03:59.420 --> 00:04:04.160
Equally well.
00:04:04.160 --> 00:04:06.250
So let's just give a
couple of examples.
00:04:06.250 --> 00:04:08.180
The first example
is the one that we
00:04:08.180 --> 00:04:12.260
did last time very laboriously.
00:04:12.260 --> 00:04:19.370
If you take the function F(x),
which happens to be x^3 / 3,
00:04:19.370 --> 00:04:24.090
then if you differentiate
it, you get, well,
00:04:24.090 --> 00:04:25.870
the the factor of 3 cancels.
00:04:25.870 --> 00:04:29.440
So you get x^2,
that's the derivative.
00:04:29.440 --> 00:04:32.760
And so by the
fundamental theorem,
00:04:32.760 --> 00:04:37.290
so this implies by the
fundamental theorem,
00:04:37.290 --> 00:04:47.140
that the integral from say, a to
b of x^3 over - sorry, x^2 dx,
00:04:47.140 --> 00:04:50.280
that's the derivative here.
00:04:50.280 --> 00:04:55.620
This is the function we're
going to use as f(x) here -
00:04:55.620 --> 00:05:02.530
is equal to this function
here, F(b) - F(a), that's here.
00:05:02.530 --> 00:05:04.220
This function here.
00:05:04.220 --> 00:05:13.470
So that's F(b) - F(a), and
that's equal to b^3 / 3 -
00:05:13.470 --> 00:05:19.480
a^3 / 3.
00:05:19.480 --> 00:05:23.710
Now, in this new
notation, we usually
00:05:23.710 --> 00:05:25.060
don't have all of these letters.
00:05:25.060 --> 00:05:26.310
All we write is the following.
00:05:26.310 --> 00:05:27.970
We write the
integral from a to b,
00:05:27.970 --> 00:05:29.470
and I'm going to
do the case 0 to b,
00:05:29.470 --> 00:05:31.803
because that was the one that
we actually did last time.
00:05:31.803 --> 00:05:35.940
So I'm going to set a = 0 here.
00:05:35.940 --> 00:05:39.590
And then, the problem we
were faced last time as this.
00:05:39.590 --> 00:05:41.940
And as I said we did
it very laboriously.
00:05:41.940 --> 00:05:47.620
But now you can see that we
can do it in ten seconds,
00:05:47.620 --> 00:05:48.280
let's say.
00:05:48.280 --> 00:05:52.300
Well, the antiderivative
of this is x^3 / 3.
00:05:52.300 --> 00:05:55.750
I'm going to evaluate it
at 0 and at b and subtract.
00:05:55.750 --> 00:06:00.540
So that's going to
be b^3 / 3 - 0^3 / 3.
00:06:00.540 --> 00:06:03.140
Which of course is b^3 / 3.
00:06:03.140 --> 00:06:06.010
And that's the end,
that's the answer.
00:06:06.010 --> 00:06:08.820
So this is a lot
faster than yesterday.
00:06:08.820 --> 00:06:10.860
I hope you'll agree.
00:06:10.860 --> 00:06:15.060
And we can dispense with
those elaborate computations.
00:06:15.060 --> 00:06:18.460
Although there's a conceptual
reason, a very important one,
00:06:18.460 --> 00:06:21.960
for understanding the
procedure that we went through.
00:06:21.960 --> 00:06:26.829
Because eventually you're
going to be using integrals
00:06:26.829 --> 00:06:28.370
and these quick ways
of doing things,
00:06:28.370 --> 00:06:32.280
to solve problems like finding
the volumes of pyramids.
00:06:32.280 --> 00:06:34.700
In other words, we're going
to reverse the process.
00:06:34.700 --> 00:06:42.100
And so we need to understand
the connection between the two.
00:06:42.100 --> 00:06:45.210
I'm going to give a
couple more examples.
00:06:45.210 --> 00:06:47.280
And then we'll go on.
00:06:47.280 --> 00:06:49.700
So the second
example would be one
00:06:49.700 --> 00:06:52.440
that would be quite difficult
to do by this Riemann sum
00:06:52.440 --> 00:06:55.310
technique that we
described yesterday.
00:06:55.310 --> 00:06:57.100
Although it is possible.
00:06:57.100 --> 00:06:59.860
It uses much higher
mathematics to do it.
00:06:59.860 --> 00:07:16.360
And that is the area under one
hump of the sine curve, sin x.
00:07:16.360 --> 00:07:17.920
Let me just draw
a picture of that.
00:07:17.920 --> 00:07:20.990
The curve goes like this, and
we're talking about this area
00:07:20.990 --> 00:07:21.490
here.
00:07:21.490 --> 00:07:24.370
It starts out at
0, it goes to pi.
00:07:24.370 --> 00:07:28.320
That's one hump.
00:07:28.320 --> 00:07:33.710
And so the answer is, it's the
integral from 0 to pi of sin
00:07:33.710 --> 00:07:37.849
x dx.
00:07:37.849 --> 00:07:39.890
And so I need to take the
antiderivative of that.
00:07:39.890 --> 00:07:42.970
And that's -cos x.
00:07:42.970 --> 00:07:46.650
That's the thing whose
derivative is sin x.
00:07:46.650 --> 00:07:49.820
Evaluating it at 0 and pi.
00:07:49.820 --> 00:07:52.070
Now, let's do this
one carefully.
00:07:52.070 --> 00:07:55.300
Because this is where I see
a lot of arithmetic mistakes.
00:07:55.300 --> 00:07:57.690
Even though this is the
easy part of the problem.
00:07:57.690 --> 00:08:02.100
It's hard to pay attention
and plug in the right numbers.
00:08:02.100 --> 00:08:04.150
And so, let's just pay
very close attention.
00:08:04.150 --> 00:08:05.530
I'm plugging in pi.
00:08:05.530 --> 00:08:08.444
That's -cos pi.
00:08:08.444 --> 00:08:09.360
That's the first term.
00:08:09.360 --> 00:08:12.240
And then I'm
subtracting the value
00:08:12.240 --> 00:08:19.980
at the bottom, which is -cos 0.
00:08:19.980 --> 00:08:22.350
There are already five
opportunities for you
00:08:22.350 --> 00:08:24.680
to make a transcription
error or an arithmetic
00:08:24.680 --> 00:08:26.740
mistake in what I just did.
00:08:26.740 --> 00:08:30.090
And I've seen all five of them.
00:08:30.090 --> 00:08:34.670
So the next one is
that this is -(-1).
00:08:34.670 --> 00:08:36.810
Minus negative 1, if you like.
00:08:36.810 --> 00:08:41.270
And then this is minus,
and here's another -1.
00:08:41.270 --> 00:08:44.250
So altogether we have 2.
00:08:44.250 --> 00:08:44.920
So that's it.
00:08:44.920 --> 00:08:46.810
That's the area.
00:08:46.810 --> 00:09:02.960
This area, which is hard
to guess, this is area 2.
00:09:02.960 --> 00:09:06.020
The third example
is maybe superfluous
00:09:06.020 --> 00:09:10.280
but I'm going to say it anyway.
00:09:10.280 --> 00:09:17.740
We can take the integral,
say, from 0 to 1, of x^100.
00:09:17.740 --> 00:09:21.260
Any power, now, is
within our power.
00:09:21.260 --> 00:09:24.050
So let's do it.
00:09:24.050 --> 00:09:32.980
So here we have the
antiderivative is x^101 / 101,
00:09:32.980 --> 00:09:36.550
evaluated at 0 and 1.
00:09:36.550 --> 00:09:42.050
And that is just 1 / 101.
00:09:42.050 --> 00:09:46.520
That's that.
00:09:46.520 --> 00:09:49.770
So that's the
fundamental theorem.
00:09:49.770 --> 00:09:53.670
Now this, as I say,
harnesses a lot
00:09:53.670 --> 00:09:58.110
of what we've already learned,
all about antiderivatives.
00:09:58.110 --> 00:10:05.820
Now, I want to give you an
intuitive interpretation.
00:10:05.820 --> 00:10:10.120
So let's try that.
00:10:10.120 --> 00:10:12.450
We'll talk about a proof
of the fundamental theorem
00:10:12.450 --> 00:10:14.170
a little bit later.
00:10:14.170 --> 00:10:16.210
It's not actually that hard.
00:10:16.210 --> 00:10:22.470
But we'll give an intuitive
reason, interpretation,
00:10:22.470 --> 00:10:28.040
if you like.
00:10:28.040 --> 00:10:37.670
Of the fundamental theorem.
00:10:37.670 --> 00:10:40.330
So this is going to
be one which is not
00:10:40.330 --> 00:10:43.990
related to area, but rather
to time and distance.
00:10:43.990 --> 00:10:55.600
So we'll consider x(t) is
your position at time t.
00:10:55.600 --> 00:11:04.020
And then x'(t), which is dx/dt,
is going to be what we know
00:11:04.020 --> 00:11:12.230
as your speed.
00:11:12.230 --> 00:11:18.150
And then what the theorem is
telling us is the following.
00:11:18.150 --> 00:11:25.580
It's telling us the integral
from a to b of v(t) dt -
00:11:25.580 --> 00:11:31.040
so, reading the relationship
- is equal to x (b) - x(a).
00:11:35.930 --> 00:11:40.800
And so this is some
kind of cumulative sum
00:11:40.800 --> 00:11:45.110
of your velocities.
00:11:45.110 --> 00:11:48.440
So let's interpret the
right-hand side first.
00:11:48.440 --> 00:11:57.290
This is the distance traveled.
00:11:57.290 --> 00:12:03.030
And it's also what you
would read on your odometer.
00:12:03.030 --> 00:12:05.550
Right, from the beginning
to the end of the trip.
00:12:05.550 --> 00:12:07.540
That's what you would
read on your odometer.
00:12:07.540 --> 00:12:19.500
Whereas this is what you would
read on your speedometer.
00:12:19.500 --> 00:12:23.210
So this is the interpretation.
00:12:23.210 --> 00:12:25.540
Now, I want to just
go one step further
00:12:25.540 --> 00:12:27.160
into this
interpretation, to make
00:12:27.160 --> 00:12:32.890
the connection with the Riemann
sums that we had yesterday.
00:12:32.890 --> 00:12:35.447
Because those are very
complicated to understand.
00:12:35.447 --> 00:12:37.280
And I want you to
understand them viscerally
00:12:37.280 --> 00:12:39.090
on several different levels.
00:12:39.090 --> 00:12:43.280
Because that's how you'll
understand integration better.
00:12:43.280 --> 00:12:44.960
The first thing that
I want to imagine,
00:12:44.960 --> 00:12:46.876
so we're going to do a
thought experiment now,
00:12:46.876 --> 00:12:50.090
which is that you are
extremely obsessive.
00:12:50.090 --> 00:12:53.410
And you're driving
your car from time a
00:12:53.410 --> 00:12:58.010
to time b, place Q
to place R, whatever.
00:12:58.010 --> 00:13:03.900
And you check your
speedometer every second.
00:13:03.900 --> 00:13:09.280
OK, so you've read your
speedometer in the i-th second,
00:13:09.280 --> 00:13:12.620
and you've read that
you're going at this speed.
00:13:12.620 --> 00:13:16.790
Now, how far do you
go in that second?
00:13:16.790 --> 00:13:19.530
Well, the answer is
you go this speed
00:13:19.530 --> 00:13:22.290
times the time interval,
which in this case
00:13:22.290 --> 00:13:24.950
we're imagining as 1 second.
00:13:24.950 --> 00:13:25.940
All right?
00:13:25.940 --> 00:13:27.980
So this is how far you went.
00:13:27.980 --> 00:13:29.230
But this is the time interval.
00:13:29.230 --> 00:13:37.070
And this is the
distance traveled
00:13:37.070 --> 00:13:46.210
in that-- second number
i, in the i-th second.
00:13:46.210 --> 00:13:48.210
The distance traveled in
the i-th second, that's
00:13:48.210 --> 00:13:49.640
a total distance you traveled.
00:13:49.640 --> 00:13:53.240
Now, what happens if you
go the whole distance?
00:13:53.240 --> 00:13:56.860
Well, you travel the sum
of all these distances.
00:13:56.860 --> 00:14:00.470
So it's some massive sum, where
n is some ridiculous number
00:14:00.470 --> 00:14:01.730
of seconds.
00:14:01.730 --> 00:14:04.250
3600 seconds or
something like that.
00:14:04.250 --> 00:14:05.070
Whatever it is.
00:14:05.070 --> 00:14:09.170
And that's going to turn out
to be very similar to what you
00:14:09.170 --> 00:14:11.770
would read on your odometer.
00:14:11.770 --> 00:14:14.140
Because during that second,
you didn't change velocity
00:14:14.140 --> 00:14:14.990
very much.
00:14:14.990 --> 00:14:17.470
So the approximation
that the speed at one
00:14:17.470 --> 00:14:21.360
time that you spotted it is
very similar to the speed
00:14:21.360 --> 00:14:22.790
during the whole second.
00:14:22.790 --> 00:14:24.430
It doesn't change that much.
00:14:24.430 --> 00:14:26.250
So this is a pretty
good approximation
00:14:26.250 --> 00:14:29.160
to how far you traveled.
00:14:29.160 --> 00:14:33.280
And so the sum is a very
realistic approximation
00:14:33.280 --> 00:14:34.810
to the entire integral.
00:14:34.810 --> 00:14:37.664
Which is denoted this way.
00:14:37.664 --> 00:14:39.080
Which, by the
fundamental theorem,
00:14:39.080 --> 00:14:43.370
is exactly how far you traveled.
00:14:43.370 --> 00:14:49.760
So this is x(b) - x(a) Exactly.
00:14:49.760 --> 00:14:55.560
The other one is approximate.
00:14:55.560 --> 00:15:08.950
OK, again this is
called a Riemann sum.
00:15:08.950 --> 00:15:17.470
All right, so that's the intro
to the fundamental theorem.
00:15:17.470 --> 00:15:23.900
And now what I need to do
is extend it just a bit.
00:15:23.900 --> 00:15:29.170
And the way I'm going to
extend it is the following.
00:15:29.170 --> 00:15:31.140
I'm going to do it on
this example first.
00:15:31.140 --> 00:15:35.530
And then we'll do
it more formally.
00:15:35.530 --> 00:15:39.200
So here's this example
where we went someplace.
00:15:39.200 --> 00:15:44.370
But now I just want to draw
you an additional picture here.
00:15:44.370 --> 00:15:49.360
Imagine I start here
and I go over to there
00:15:49.360 --> 00:15:54.650
and then I come back.
00:15:54.650 --> 00:15:56.100
And maybe even I
do a round trip.
00:15:56.100 --> 00:15:58.090
I come back to the same place.
00:15:58.090 --> 00:16:01.070
Well, if I come back
to the same place,
00:16:01.070 --> 00:16:06.120
then the position is unchanged
from the beginning to the end.
00:16:06.120 --> 00:16:08.140
In other words, the
difference is 0.
00:16:08.140 --> 00:16:12.632
And the velocity, technically
rather than the speed.
00:16:12.632 --> 00:16:14.840
It's the speed to the right
and the speed to the left
00:16:14.840 --> 00:16:16.640
maybe are the same,
but one of them
00:16:16.640 --> 00:16:18.330
is going in the positive
direction and one of them
00:16:18.330 --> 00:16:19.788
is going in the
negative direction,
00:16:19.788 --> 00:16:22.090
and they cancel each other.
00:16:22.090 --> 00:16:25.340
So if you have this
kind of situation,
00:16:25.340 --> 00:16:26.940
we want that to be reflected.
00:16:26.940 --> 00:16:28.550
We like that
interpretation and we
00:16:28.550 --> 00:16:32.260
want to preserve it even
when-- in the case when
00:16:32.260 --> 00:16:35.090
the function v is negative.
00:16:35.090 --> 00:16:47.280
And so I'm going to now extend
our notion of integration.
00:16:47.280 --> 00:17:02.320
So we'll extend integration
to the case f negative.
00:17:02.320 --> 00:17:04.420
Or positive.
00:17:04.420 --> 00:17:08.750
In other words, it
could be any sign.
00:17:08.750 --> 00:17:10.550
Actually, there's no change.
00:17:10.550 --> 00:17:12.430
The formulas are all the same.
00:17:12.430 --> 00:17:14.891
We just-- If this v is
going to be positive,
00:17:14.891 --> 00:17:16.140
we write in a positive number.
00:17:16.140 --> 00:17:18.640
If it's going to be negative,
we write in a negative number.
00:17:18.640 --> 00:17:20.270
And we just leave it alone.
00:17:20.270 --> 00:17:25.480
And the real-- So here's--
Let me carry out an example
00:17:25.480 --> 00:17:29.230
and show you how it works.
00:17:29.230 --> 00:17:32.750
I'll carry out the example
on this blackboard up here.
00:17:32.750 --> 00:17:33.670
Of the sine function.
00:17:33.670 --> 00:17:36.020
But we're going
to try two humps.
00:17:36.020 --> 00:17:38.690
We're going to try the
first hump and the one that
00:17:38.690 --> 00:17:39.880
goes underneath.
00:17:39.880 --> 00:17:40.380
There.
00:17:40.380 --> 00:17:43.830
So our example here is
going to be the integral
00:17:43.830 --> 00:17:50.910
from 0 to 2pi of sin x dx.
00:17:50.910 --> 00:17:55.710
And now, because the fundamental
theorem is so important, and so
00:17:55.710 --> 00:17:58.410
useful, and so
convenient, we just
00:17:58.410 --> 00:18:01.100
assume that it be true
in this case as well.
00:18:01.100 --> 00:18:07.170
So we insist that this is going
to be -cos x, evaluated at 0
00:18:07.170 --> 00:18:10.540
and 2pi, with the difference.
00:18:10.540 --> 00:18:12.770
Now, when we carry
out that difference,
00:18:12.770 --> 00:18:19.320
what we get here is
-cos 2pi - (-cos 0).
00:18:24.670 --> 00:18:33.670
Which is -1 - (-1), which is 0.
00:18:33.670 --> 00:18:39.650
And the interpretation
of this is the following.
00:18:39.650 --> 00:18:43.840
Here's our double hump,
here's pi and here's 2pi.
00:18:43.840 --> 00:18:47.590
And all that's happening is that
the geometric interpretation
00:18:47.590 --> 00:18:50.440
that we had before of
the area under the curve
00:18:50.440 --> 00:18:53.320
has to be taken with a grain
of salt. In other words,
00:18:53.320 --> 00:18:56.530
I lied to you before when I said
that the definite integral was
00:18:56.530 --> 00:18:57.610
the area under the curve.
00:18:57.610 --> 00:18:59.030
It's not.
00:18:59.030 --> 00:19:00.790
The definite
integral is the area
00:19:00.790 --> 00:19:03.470
under the curve when
it's above the curve,
00:19:03.470 --> 00:19:08.500
and it counts negatively
when it's below the curve.
00:19:08.500 --> 00:19:12.690
So yesterday, my geometric
interpretation was incomplete.
00:19:12.690 --> 00:19:19.180
And really just a plain lie.
00:19:19.180 --> 00:19:28.860
So the true geometric
interpretation
00:19:28.860 --> 00:19:43.760
of the definite integral
is plus the area
00:19:43.760 --> 00:19:49.890
above the axis,
above the x-axis,
00:19:49.890 --> 00:20:00.360
minus the area below the x-axis.
00:20:00.360 --> 00:20:02.580
As in the picture.
00:20:02.580 --> 00:20:04.230
I'm just writing
it down in words,
00:20:04.230 --> 00:20:08.870
but you should think
of it visually also.
00:20:08.870 --> 00:20:12.820
So that's the setup here.
00:20:12.820 --> 00:20:17.530
And now we have the complete
definition of integrals.
00:20:17.530 --> 00:20:19.950
And I need to list for you
a bunch of their properties
00:20:19.950 --> 00:20:21.740
and how we deal with integrals.
00:20:21.740 --> 00:20:25.270
So are there any
questions before we go on?
00:20:25.270 --> 00:20:25.770
Yeah.
00:20:25.770 --> 00:20:32.240
STUDENT: [INAUDIBLE]
00:20:32.240 --> 00:20:39.000
PROFESSOR: Right.
00:20:39.000 --> 00:20:42.733
So the question was,
wouldn't the absolute value
00:20:42.733 --> 00:20:45.720
of the velocity
function be involved?
00:20:45.720 --> 00:20:48.300
The answer is yes.
00:20:48.300 --> 00:20:51.730
That is, that's one
question that you could ask.
00:20:51.730 --> 00:20:54.430
One question you
could ask is what's
00:20:54.430 --> 00:20:57.360
the total distance traveled.
00:20:57.360 --> 00:21:00.600
And in that case,
you would keep track
00:21:00.600 --> 00:21:05.534
of the absolute value of
the velocity as you said,
00:21:05.534 --> 00:21:06.950
whether it's
positive or negative.
00:21:06.950 --> 00:21:13.970
And then you would get the
total length of this curve here.
00:21:13.970 --> 00:21:18.520
That's, however, not what the
definite integral measures.
00:21:18.520 --> 00:21:21.640
It measures the net
distance traveled.
00:21:21.640 --> 00:21:24.049
So it's another thing.
00:21:24.049 --> 00:21:25.340
In other words, we can do that.
00:21:25.340 --> 00:21:27.890
We now have the
tools to do both.
00:21:27.890 --> 00:21:35.180
We could also-- So if you
like, the total distance
00:21:35.180 --> 00:21:39.550
is equal to the
integral of this.
00:21:39.550 --> 00:21:40.730
From a to b.
00:21:40.730 --> 00:21:49.200
But the net distance is the
one without the absolute value
00:21:49.200 --> 00:21:54.620
signs.
00:21:54.620 --> 00:21:57.350
So that's correct.
00:21:57.350 --> 00:22:03.950
Other questions?
00:22:03.950 --> 00:22:04.590
All right.
00:22:04.590 --> 00:22:23.950
So now, let's talk about
properties of integrals.
00:22:23.950 --> 00:22:37.630
So the properties of integrals
that I want to mention to you
00:22:37.630 --> 00:22:39.030
are these.
00:22:39.030 --> 00:22:47.280
The first one doesn't
bear too much comment.
00:22:47.280 --> 00:22:53.370
If you take the cumulative
integral of a sum,
00:22:53.370 --> 00:22:58.820
you're just trying to get the
sum of the separate integrals
00:22:58.820 --> 00:23:01.867
here.
00:23:01.867 --> 00:23:03.200
And I won't say much about that.
00:23:03.200 --> 00:23:06.660
That's because sums come
out, the because the integral
00:23:06.660 --> 00:23:07.790
is a sum.
00:23:07.790 --> 00:23:15.330
Incidentally, you know
this strange symbol here,
00:23:15.330 --> 00:23:17.302
there's actually a reason
for it historically.
00:23:17.302 --> 00:23:18.760
If you go back to
old books, you'll
00:23:18.760 --> 00:23:21.880
see that it actually looks
a little bit more like an S.
00:23:21.880 --> 00:23:24.530
This capital sigma is a sum.
00:23:24.530 --> 00:23:27.170
S for sum, because
everybody in those days
00:23:27.170 --> 00:23:28.400
knew Latin and Greek.
00:23:28.400 --> 00:23:31.360
And this one is also
an S, but gradually it
00:23:31.360 --> 00:23:33.390
was such an important S
that they made a bigger.
00:23:33.390 --> 00:23:35.890
And then they stretched it out
and made it a little thinner,
00:23:35.890 --> 00:23:40.590
because it didn't fit into
one typesetting space.
00:23:40.590 --> 00:23:43.550
And so just for typesetting
reasons it got stretched.
00:23:43.550 --> 00:23:45.390
And got a little bit skinny.
00:23:45.390 --> 00:23:48.640
Anyway, so it's really an
S. And in fact, in French
00:23:48.640 --> 00:23:50.800
they call it sum.
00:23:50.800 --> 00:23:53.900
Even though we call it integral.
00:23:53.900 --> 00:23:57.690
So it's a sum.
00:23:57.690 --> 00:24:00.450
So it's consistent
with sums in this way.
00:24:00.450 --> 00:24:09.040
And similarly, similarly we can
factor constants out of sums.
00:24:09.040 --> 00:24:20.820
So if you have an integral like
this, the constant factors out.
00:24:20.820 --> 00:24:25.140
But definitely don't try to
get a function out of this.
00:24:25.140 --> 00:24:27.410
That won't happen.
00:24:27.410 --> 00:24:30.470
OK, in other words, c
has to be a constant.
00:24:30.470 --> 00:24:41.320
Doesn't depend on x.
00:24:41.320 --> 00:24:44.340
The third property.
00:24:44.340 --> 00:24:46.710
What do I want to call
the third property here?
00:24:46.710 --> 00:24:51.522
I have sort of a preliminary
property, yes, here.
00:24:51.522 --> 00:24:52.480
Which is the following.
00:24:52.480 --> 00:24:53.730
And I'll draw a picture of it.
00:24:53.730 --> 00:24:59.410
I suppose you have three
points along a line.
00:24:59.410 --> 00:25:01.300
So then I'm going to
draw a picture of that.
00:25:01.300 --> 00:25:03.940
And I'm going to use the
interpretation above the curve,
00:25:03.940 --> 00:25:05.564
even though that's
not the whole thing.
00:25:05.564 --> 00:25:07.990
So here's a, here's
b and here's c.
00:25:07.990 --> 00:25:10.490
And you can see that
the area of this piece,
00:25:10.490 --> 00:25:14.080
of the first two pieces
here, when added together,
00:25:14.080 --> 00:25:15.820
gives you the area of the whole.
00:25:15.820 --> 00:25:19.400
And that's the rule that
I'd like to tell you.
00:25:19.400 --> 00:25:23.540
So if you integrate
from a to b, and you
00:25:23.540 --> 00:25:28.070
add to that the
integral from b to c,
00:25:28.070 --> 00:25:37.344
you'll get the
integral from a to c.
00:25:37.344 --> 00:25:39.260
This is going to be just
a little preliminary,
00:25:39.260 --> 00:25:41.830
because the rule is a
little better than this.
00:25:41.830 --> 00:25:47.970
But I will explain
that in a minute.
00:25:47.970 --> 00:25:52.540
The fourth rule is
a very simple one.
00:25:52.540 --> 00:25:57.910
Which is that the integral
from a to a of f(x) dx
00:25:57.910 --> 00:26:00.624
is equal to 0.
00:26:00.624 --> 00:26:02.790
Now, that you can see very
obviously because there's
00:26:02.790 --> 00:26:04.140
no area.
00:26:04.140 --> 00:26:05.900
No horizontal movement there.
00:26:05.900 --> 00:26:08.430
The rectangle is
infinitely thin,
00:26:08.430 --> 00:26:10.350
and there's nothing there.
00:26:10.350 --> 00:26:12.020
So this is the case.
00:26:12.020 --> 00:26:17.860
You can also interpret
it a F(a) - F(a).
00:26:17.860 --> 00:26:21.860
So that's also consistent
with our interpretation.
00:26:21.860 --> 00:26:24.510
In terms of the fundamental
theorem of calculus.
00:26:24.510 --> 00:26:28.020
And it's perfectly reasonable
that this is the case.
00:26:28.020 --> 00:26:33.090
Now, the fifth property
is a definition.
00:26:33.090 --> 00:26:35.110
It's not really a property.
00:26:35.110 --> 00:26:38.080
But it's very important.
00:26:38.080 --> 00:26:46.040
The integral from a to b of f(x)
dx equal to minus the integral
00:26:46.040 --> 00:26:50.960
from b to a, of f( x) dx.
00:26:50.960 --> 00:26:57.490
Now, really, the right-hand side
here is an undefined quantity
00:26:57.490 --> 00:26:58.740
so far.
00:26:58.740 --> 00:27:02.380
We never said you
could ever do this
00:27:02.380 --> 00:27:05.650
where the a is less than the b.
00:27:05.650 --> 00:27:09.600
Because this is
working backwards here.
00:27:09.600 --> 00:27:12.232
But we just have a convention
that that's the definition.
00:27:12.232 --> 00:27:13.690
Whenever we write
down this number,
00:27:13.690 --> 00:27:17.120
it's the same as minus
what that number is.
00:27:17.120 --> 00:27:20.229
And the reason for
all of these is again
00:27:20.229 --> 00:27:22.520
that we want them to be
consistent with the fundamental
00:27:22.520 --> 00:27:23.730
theorem of calculus.
00:27:23.730 --> 00:27:26.250
Which is the thing that
makes all of this work.
00:27:26.250 --> 00:27:33.210
So if you notice the left-hand
side here is F(b) - F(a),
00:27:33.210 --> 00:27:36.410
capital F, the
antiderivative of little f.
00:27:36.410 --> 00:27:39.420
On the other hand, the
other side is minus,
00:27:39.420 --> 00:27:42.050
and if we just ignore that,
we say these are letters,
00:27:42.050 --> 00:27:44.633
if we were a machine, we didn't
know which one was bigger than
00:27:44.633 --> 00:27:49.980
which, we just plugged them in,
we would get here F(a) - F(b),
00:27:49.980 --> 00:27:50.690
over here.
00:27:50.690 --> 00:27:53.402
And to make these two
things equal, what we want
00:27:53.402 --> 00:27:54.610
is to put that minus sign in.
00:27:54.610 --> 00:27:59.620
Now it's consistent.
00:27:59.620 --> 00:28:02.610
So again, these
rules are set up so
00:28:02.610 --> 00:28:05.420
that everything is consistent.
00:28:05.420 --> 00:28:11.190
And now I want to
improve on rule 3 here.
00:28:11.190 --> 00:28:15.010
And point out to you - so
let me just go back to rule 3
00:28:15.010 --> 00:28:21.030
for a second - that now that
we can evaluate integrals
00:28:21.030 --> 00:28:24.920
regardless of the order, we
don't have to have a < b,
00:28:24.920 --> 00:28:28.070
b < c in order to make
sense out of this.
00:28:28.070 --> 00:28:31.740
We actually have the possibility
of considering integrals
00:28:31.740 --> 00:28:34.180
where the a's and the
b's and the c's are
00:28:34.180 --> 00:28:36.270
in any order you want.
00:28:36.270 --> 00:28:38.630
And in fact, with
this definition,
00:28:38.630 --> 00:28:43.140
with this definition 5, 3 works
no matter what the numbers are.
00:28:43.140 --> 00:28:44.710
So this is much more convenient.
00:28:44.710 --> 00:28:49.570
We don't, this is not necessary.
00:28:49.570 --> 00:28:51.070
Not necessary.
00:28:51.070 --> 00:28:57.220
It just works
using convention 5.
00:28:57.220 --> 00:29:04.140
OK, with 5.
00:29:04.140 --> 00:29:08.760
Again, before I go
on, let me emphasize:
00:29:08.760 --> 00:29:11.510
we really want to respect
the sign of this velocity.
00:29:11.510 --> 00:29:15.110
We really want the net
change in the position.
00:29:15.110 --> 00:29:18.180
And we don't want this
absolute value here.
00:29:18.180 --> 00:29:20.740
Because otherwise, all of our
formulas are going to mess up.
00:29:20.740 --> 00:29:22.260
We won't always
be able to check.
00:29:22.260 --> 00:29:24.955
Sometimes you have
letters rather than
00:29:24.955 --> 00:29:26.580
actual numbers here,
and you won't know
00:29:26.580 --> 00:29:28.160
whether a is bigger than b.
00:29:28.160 --> 00:29:31.370
So you'll want to know that
these formulas work and are
00:29:31.370 --> 00:29:36.410
consistent in all situations.
00:29:36.410 --> 00:29:39.550
OK, I'm going to
trade these again.
00:29:39.550 --> 00:29:47.250
In order to preserve the
ordering 1 through 5.
00:29:47.250 --> 00:29:54.200
And now I have a sixth property
that I want to talk about.
00:29:54.200 --> 00:30:02.010
This one is called estimation.
00:30:02.010 --> 00:30:05.470
And it says the following.
00:30:05.470 --> 00:30:18.270
If f(x) <= g(x), then the
integral from a to b of f(x) dx
00:30:18.270 --> 00:30:23.180
is less than or equal to the
integral from a to b of g(x)
00:30:23.180 --> 00:30:28.300
dx.
00:30:28.300 --> 00:30:36.860
Now, this one says that if I'm
going more slowly than you,
00:30:36.860 --> 00:30:40.840
then you go farther than I do.
00:30:40.840 --> 00:30:41.760
OK.
00:30:41.760 --> 00:30:43.710
That's all it's saying.
00:30:43.710 --> 00:30:47.870
For this one, you'd
better have a < b.
00:30:47.870 --> 00:30:49.100
You need it.
00:30:49.100 --> 00:30:55.360
Because we flip the signs when
we flip the order of a and b.
00:30:55.360 --> 00:30:59.410
So this one, it's essential
that the lower limit be smaller
00:30:59.410 --> 00:31:04.600
than the upper limit.
00:31:04.600 --> 00:31:06.930
But let me just emphasize,
because we're dealing
00:31:06.930 --> 00:31:08.370
with the generalities of this.
00:31:08.370 --> 00:31:10.350
Actually if one of
these is negative
00:31:10.350 --> 00:31:14.650
and the other one is
negative, then it also works.
00:31:14.650 --> 00:31:17.510
This one ends up being, if
f is more negative than g,
00:31:17.510 --> 00:31:22.380
then this added up thing is
more negative than that one.
00:31:22.380 --> 00:31:25.270
Again, under the assumption
that a is less than b.
00:31:25.270 --> 00:31:34.760
So as I wrote it it's
in full generality.
00:31:34.760 --> 00:31:37.490
Let's illustrate this one.
00:31:37.490 --> 00:31:47.030
And then we have one more
property to learn after that.
00:31:47.030 --> 00:31:58.970
So let me give you an
example of estimation.
00:31:58.970 --> 00:32:01.970
The example is the same as
one that I already gave you.
00:32:01.970 --> 00:32:05.080
But this time, because we
have the tool of integration,
00:32:05.080 --> 00:32:11.290
we can just follow our
noses and it works.
00:32:11.290 --> 00:32:14.400
I start with the
inequality, so I'm
00:32:14.400 --> 00:32:16.090
trying to illustrate
estimation, so I
00:32:16.090 --> 00:32:17.673
want to start with
an inequality which
00:32:17.673 --> 00:32:19.290
is what the hypothesis is here.
00:32:19.290 --> 00:32:21.480
And I'm going to
integrate the inequality
00:32:21.480 --> 00:32:22.900
to get this conclusion.
00:32:22.900 --> 00:32:25.860
And see what conclusion it is.
00:32:25.860 --> 00:32:30.160
The inequality that I want
to take is that e^x >= 1,
00:32:30.160 --> 00:32:32.800
for x >= 0.
00:32:32.800 --> 00:32:37.620
That's going to be
our starting place.
00:32:37.620 --> 00:32:39.070
And now I'm going
to integrate it.
00:32:39.070 --> 00:32:40.850
That is, I'm going
to use estimation
00:32:40.850 --> 00:32:42.420
to see what that gives.
00:32:42.420 --> 00:32:45.100
Well, I'm going to
integrate, say, from 0 to b.
00:32:45.100 --> 00:32:50.010
I can't integrate below 0
because it's only true above 0.
00:32:50.010 --> 00:32:54.390
This is e^x dx greater than or
equal to the integral from 0
00:32:54.390 --> 00:33:01.990
to b of 1 dx.
00:33:01.990 --> 00:33:05.980
Alright, let's work out
what each of these is.
00:33:05.980 --> 00:33:14.370
The first one, e^x dx, is,
the antiderivative is e^x,
00:33:14.370 --> 00:33:16.120
evaluated at 0 and b.
00:33:16.120 --> 00:33:18.930
So that's e^b - e^0.
00:33:18.930 --> 00:33:23.510
Which is e^b - 1.
00:33:23.510 --> 00:33:28.560
The other one,
you're supposed to be
00:33:28.560 --> 00:33:32.270
able to get by
the rectangle law.
00:33:32.270 --> 00:33:35.840
This is one rectangle
of base b and height 1.
00:33:35.840 --> 00:33:37.350
So the answer is b.
00:33:37.350 --> 00:33:44.100
Or you can do it by
antiderivatives, but it's b.
00:33:44.100 --> 00:33:49.010
That means that our inequality
says if I just combine these
00:33:49.010 --> 00:33:55.640
two things together,
that e^b - 1 >= b.
00:33:55.640 --> 00:34:02.040
And that's the same
thing as e^b >= 1 + b.
00:34:02.040 --> 00:34:05.010
Again, this only
works for b >= 0.
00:34:05.010 --> 00:34:10.520
Notice that if b were
negative, this would be a well
00:34:10.520 --> 00:34:13.520
defined quantity.
00:34:13.520 --> 00:34:18.240
But this estimation
would be false.
00:34:18.240 --> 00:34:22.010
We need that the b > 0 in
order for this to make sense.
00:34:22.010 --> 00:34:24.590
So this was used.
00:34:24.590 --> 00:34:28.640
And that's a good thing, because
this inequality is suspect.
00:34:28.640 --> 00:34:32.060
Actually, it turns out to
be true when b is negative.
00:34:32.060 --> 00:34:38.830
But we certainly
didn't prove it.
00:34:38.830 --> 00:34:42.260
I'm going to just
repeat this process.
00:34:42.260 --> 00:34:46.400
So let's repeat it.
00:34:46.400 --> 00:34:49.890
Starting from the
inequality, the conclusion,
00:34:49.890 --> 00:34:51.390
which is sitting right here.
00:34:51.390 --> 00:34:59.970
But I'll write it in a form
e^x >= 1 + x, for x >= 0.
00:34:59.970 --> 00:35:02.160
And now, if I
integrate this one,
00:35:02.160 --> 00:35:06.245
I get the integral from 0 to b,
e^x dx is greater than or equal
00:35:06.245 --> 00:35:12.360
to the integral from
0 to b, (1 + x) dx,
00:35:12.360 --> 00:35:16.020
and I remind you that we've
already calculated this one.
00:35:16.020 --> 00:35:19.000
This is e^b - 1.
00:35:19.000 --> 00:35:21.710
And the other one is
not hard to calculate.
00:35:21.710 --> 00:35:25.930
The antiderivative
is x + x^2 / 2.
00:35:25.930 --> 00:35:28.690
We're evaluating
that at 0 and b.
00:35:28.690 --> 00:35:34.830
So that comes out
to be b + b^2 / 2.
00:35:34.830 --> 00:35:43.550
And so our conclusion is that
the left side, which is e^b -
00:35:43.550 --> 00:35:48.300
1 >= b + b^2 / 2.
00:35:48.300 --> 00:35:51.930
And this is for b >= 0.
00:35:51.930 --> 00:36:00.510
And that's the same thing
as e^b >= 1 + b + b^2 / 2.
00:36:00.510 --> 00:36:04.050
This one actually is
false for b negative,
00:36:04.050 --> 00:36:09.280
so that's something
that you have
00:36:09.280 --> 00:36:15.560
to be careful with
the b positive's here.
00:36:15.560 --> 00:36:17.350
So you can keep on
going with this,
00:36:17.350 --> 00:36:20.360
and you didn't have to think.
00:36:20.360 --> 00:36:23.330
And you'll produce a very
interesting polynomial,
00:36:23.330 --> 00:36:25.340
which is a good
approximation to e^b.
00:36:30.720 --> 00:36:34.520
So that's it for the
basic properties.
00:36:34.520 --> 00:36:38.980
Now there's one tricky property
that I need to tell you about.
00:36:38.980 --> 00:36:47.920
It's not that tricky,
but it's a little tricky.
00:36:47.920 --> 00:37:07.390
And this is change of variables.
00:37:07.390 --> 00:37:09.600
Change of variables
in integration,
00:37:09.600 --> 00:37:11.220
we've actually already done.
00:37:11.220 --> 00:37:14.190
We called that, the last
time we talked about it,
00:37:14.190 --> 00:37:23.430
we called it substitution.
00:37:23.430 --> 00:37:26.210
And the idea here,
if you may remember,
00:37:26.210 --> 00:37:31.550
was that if you're faced
with an integral like this,
00:37:31.550 --> 00:37:38.340
you can change it to, if you put
in u = u(x) and you have a du,
00:37:38.340 --> 00:37:42.700
which is equal to
u'(x) du-- dx, sorry.
00:37:42.700 --> 00:37:45.900
Then you can change the
integral as follows.
00:37:45.900 --> 00:37:51.630
This is the same as
g(u(x)) u'(x) dx.
00:37:51.630 --> 00:37:58.920
This was the general
procedure for substitution.
00:37:58.920 --> 00:38:05.500
What's new today is that we're
going to put in the limits.
00:38:05.500 --> 00:38:10.350
If you have a limit here,
u_1, and a limit here, u_2,
00:38:10.350 --> 00:38:12.790
you want to know what
the relationship is
00:38:12.790 --> 00:38:15.690
between the limits here and
the limits when you change
00:38:15.690 --> 00:38:18.780
variables to the new variables.
00:38:18.780 --> 00:38:21.290
And it's the simplest
possible thing.
00:38:21.290 --> 00:38:25.760
Namely the two limits over here
are in the same relationship
00:38:25.760 --> 00:38:29.130
as u(x) is to this
symbol u here.
00:38:29.130 --> 00:38:35.830
In other words, u_1 =
u(x_1), and u_2 = u(x_2).
00:38:35.830 --> 00:38:39.310
That's what works.
00:38:39.310 --> 00:38:42.620
Now there's only
one danger here,
00:38:42.620 --> 00:38:48.230
there's one subtlety
which is, this only works
00:38:48.230 --> 00:39:02.010
if u' does not change sign.
00:39:02.010 --> 00:39:04.680
I've been worrying a little
bit about going backwards
00:39:04.680 --> 00:39:06.400
and forwards, and
I allowed myself
00:39:06.400 --> 00:39:08.525
to reverse and do all
kinds of stuff, right,
00:39:08.525 --> 00:39:09.400
with these integrals.
00:39:09.400 --> 00:39:11.560
So we're sort of free to do it.
00:39:11.560 --> 00:39:14.560
Well, this is one case where
you want to avoid it, OK?
00:39:14.560 --> 00:39:15.850
Just don't do it.
00:39:15.850 --> 00:39:17.990
It is possible, actually,
to make sense out of it,
00:39:17.990 --> 00:39:21.810
but it's also possible to get
yourself infinitely confused.
00:39:21.810 --> 00:39:24.350
So just make sure
that-- Now, it's
00:39:24.350 --> 00:39:27.580
OK if u' is always negative,
or always going one way,
00:39:27.580 --> 00:39:30.034
so OK if u' is always
positive, you're always
00:39:30.034 --> 00:39:31.700
going the other way,
but if you mix them
00:39:31.700 --> 00:39:39.960
up you'll get yourself mixed up.
00:39:39.960 --> 00:39:46.210
Let me give you an example.
00:39:46.210 --> 00:39:54.230
The example will be maybe
close to what we did last time.
00:39:54.230 --> 00:39:57.900
When we first did
substitution, I mean.
00:39:57.900 --> 00:40:02.160
So the integral from 1 to 2,
this time I'll put in definite
00:40:02.160 --> 00:40:09.690
limits, of x^2 plus-- sorry,
maybe I call this x^3. x^3 + 2,
00:40:09.690 --> 00:40:17.950
let's say, I don't know,
to the 5th power, x^2 dx.
00:40:17.950 --> 00:40:20.310
So this is an example
of an integral
00:40:20.310 --> 00:40:25.900
that we would have tried to
handle by substitution before.
00:40:25.900 --> 00:40:36.590
And the substitution we would
have used is u = x^3 + 2.
00:40:36.590 --> 00:40:38.560
And that's exactly what
we're going to do here.
00:40:38.560 --> 00:40:44.710
But we're just going to also
take into account the limits.
00:40:44.710 --> 00:40:47.730
The first step, as in any
substitution or change
00:40:47.730 --> 00:40:54.070
of variables, is this.
00:40:54.070 --> 00:40:57.056
And so we can fill
in the things that we
00:40:57.056 --> 00:40:58.180
would have done previously.
00:40:58.180 --> 00:41:01.790
Which is that this is the
integral and this is u^5.
00:41:01.790 --> 00:41:09.300
And then because this is
3x^2, we see that this is 3.
00:41:09.300 --> 00:41:13.130
Sorry, let's write
it the other way.
00:41:13.130 --> 00:41:17.480
1/3 du = x^2 dx.
00:41:17.480 --> 00:41:20.370
So that's what I'm going to
plug in for this factor here.
00:41:20.370 --> 00:41:26.480
So here's 1/3 du,
which replaces that.
00:41:26.480 --> 00:41:29.430
But now there's
the extra feature.
00:41:29.430 --> 00:41:31.660
The extra feature is the limits.
00:41:31.660 --> 00:41:35.160
So here, really in
disguise, because, and now
00:41:35.160 --> 00:41:38.280
this is incredibly important.
00:41:38.280 --> 00:41:44.070
This is one of the reasons why
we use this notation dx and du.
00:41:44.070 --> 00:41:47.120
We want to remind
ourselves which variable
00:41:47.120 --> 00:41:49.170
is involved in the integration.
00:41:49.170 --> 00:41:52.470
And especially if you're the
one naming the variables,
00:41:52.470 --> 00:41:54.760
you may get mixed
up in this respect.
00:41:54.760 --> 00:41:59.200
So you must know which variable
is varying between 1 and 2.
00:41:59.200 --> 00:42:01.510
And the answer is, it's
x is the one that's
00:42:01.510 --> 00:42:04.150
varying between 1 and 2.
00:42:04.150 --> 00:42:06.870
So in disguise, even
though I didn't write it,
00:42:06.870 --> 00:42:10.260
it was contained in
this little symbol here.
00:42:10.260 --> 00:42:11.840
This reminded us which variable.
00:42:11.840 --> 00:42:14.140
You'll find this amazingly
important when you
00:42:14.140 --> 00:42:16.480
get to multivariable calculus.
00:42:16.480 --> 00:42:18.810
When there are many
variables floating around.
00:42:18.810 --> 00:42:21.390
So this is an incredibly
important distinction to make.
00:42:21.390 --> 00:42:23.300
So now, over here
we have a limit.
00:42:23.300 --> 00:42:26.270
But of course it's supposed
to be with respect to u, now.
00:42:26.270 --> 00:42:29.490
So we need to calculate what
those corresponding limits are.
00:42:29.490 --> 00:42:33.270
And indeed it's just, I plug in
here u_1 is going to be equal
00:42:33.270 --> 00:42:37.430
to what I plug in for x = 1,
that's going to be 1^3 + 2,
00:42:37.430 --> 00:42:38.760
which is 3.
00:42:38.760 --> 00:42:46.390
And then u_2 is 2^3 + 2,
which is equal to 10, right?
00:42:46.390 --> 00:42:47.960
8 + 2 = 10.
00:42:47.960 --> 00:42:57.700
So this is the integral
from 3 to 10, of u^5 1/3 du.
00:42:57.700 --> 00:43:00.380
And now I can
finish the problem.
00:43:00.380 --> 00:43:06.390
This is 1/18 u^6, from 3 to 10.
00:43:06.390 --> 00:43:10.000
And this is where the
most common mistake occurs
00:43:10.000 --> 00:43:12.360
in substitutions of this type.
00:43:12.360 --> 00:43:14.850
Which is that if
you ignore this,
00:43:14.850 --> 00:43:17.020
and you plug in
these 1 and 2 here,
00:43:17.020 --> 00:43:20.090
you think, oh I should just
be putting it at 1 and 2.
00:43:20.090 --> 00:43:22.940
But actually, it
should be, the u-value
00:43:22.940 --> 00:43:26.400
that we're interested in,
and the lower limit is u = 3
00:43:26.400 --> 00:43:29.660
and u = 10 is the upper limit.
00:43:29.660 --> 00:43:31.405
So those are suppressed here.
00:43:31.405 --> 00:43:35.620
But those are the
ones that we want.
00:43:35.620 --> 00:43:37.360
And so, here we go.
00:43:37.360 --> 00:43:41.770
It's 1/18 times some ridiculous
number which I won't calculate.
00:43:41.770 --> 00:43:44.260
10^6 - - 3^6.
00:43:47.820 --> 00:43:48.630
Yes, question.
00:43:48.630 --> 00:44:07.250
STUDENT: [INAUDIBLE]
00:44:07.250 --> 00:44:10.380
PROFESSOR: So, if
you want to do things
00:44:10.380 --> 00:44:16.290
with where you're worrying
about the sign change,
00:44:16.290 --> 00:44:20.820
the right strategy is,
what you suggested works.
00:44:20.820 --> 00:44:23.090
And in fact I'm going to
do an example right now
00:44:23.090 --> 00:44:24.220
on this subject.
00:44:24.220 --> 00:44:30.160
But, the right strategy is
to break it up into pieces.
00:44:30.160 --> 00:44:34.950
Where u' has one sign
or the other, OK?
00:44:34.950 --> 00:44:37.430
Let me show you an example.
00:44:37.430 --> 00:44:40.700
Where things go wrong.
00:44:40.700 --> 00:44:47.900
And I'll tell you how
to handle it, roughly.
00:44:47.900 --> 00:44:55.790
So here's our warning.
00:44:55.790 --> 00:45:00.280
Suppose you're integrating
from -1 to 1, x^2 dx.
00:45:00.280 --> 00:45:02.810
Here's an example.
00:45:02.810 --> 00:45:09.654
And you have the temptation
to plug in u = x^2.
00:45:09.654 --> 00:45:11.570
Now, of course, we know
how to integrate this.
00:45:11.570 --> 00:45:16.390
But let's just pretend we
were stubborn and wanted
00:45:16.390 --> 00:45:19.260
to use substitution.
00:45:19.260 --> 00:45:26.610
Then we have du = 2x dx.
00:45:26.610 --> 00:45:30.170
And now if I try to
make the correspondence,
00:45:30.170 --> 00:45:37.130
notice that the limits
are u_1 = (-1)^2,
00:45:37.130 --> 00:45:38.860
that's the bottom limit.
00:45:38.860 --> 00:45:40.830
And u_2 is the upper limit.
00:45:40.830 --> 00:45:43.450
That's 1^2, that's
also equal to 1.
00:45:43.450 --> 00:45:45.250
Both limits are 1.
00:45:45.250 --> 00:45:47.750
So this is going from 1 to 1.
00:45:47.750 --> 00:45:53.350
And no matter what it is,
we know it's going to be 0.
00:45:53.350 --> 00:45:55.610
But we know this is not 0.
00:45:55.610 --> 00:45:58.620
This is the integral
of a positive quantity.
00:45:58.620 --> 00:46:03.139
And the area under a curve is
going to be a positive area.
00:46:03.139 --> 00:46:04.430
So this is a positive quantity.
00:46:04.430 --> 00:46:07.300
It can't be 0.
00:46:07.300 --> 00:46:12.090
If you actually plug it in,
it looks equally strange.
00:46:12.090 --> 00:46:16.160
You put in here this u and then,
so that would be for the u^2.
00:46:16.160 --> 00:46:22.110
And then to plug in for dx,
you would write dx = 1/(2x) du.
00:46:22.110 --> 00:46:27.760
And then you might
write that as this.
00:46:27.760 --> 00:46:31.500
And so what I should put in
here is this quantity here.
00:46:31.500 --> 00:46:33.360
Which is a perfectly
OK integral.
00:46:33.360 --> 00:46:37.870
And it has a value, I
mean, it's what it is.
00:46:37.870 --> 00:46:39.140
It's 0.
00:46:39.140 --> 00:46:45.270
So of course this is not true.
00:46:45.270 --> 00:46:51.540
And the reason is that
u was equal to x^2,
00:46:51.540 --> 00:46:57.340
and u'(x) was equal to 2x, which
was positive for x positive,
00:46:57.340 --> 00:47:00.070
and negative for x negative.
00:47:00.070 --> 00:47:03.470
And this was the sign change
which causes us trouble.
00:47:03.470 --> 00:47:08.160
If we break it off into its
two halves, then it'll be OK
00:47:08.160 --> 00:47:09.710
and you'll be able to use this.
00:47:09.710 --> 00:47:12.120
Now, there was a mistake.
00:47:12.120 --> 00:47:15.540
And this was essentially
what you were saying.
00:47:15.540 --> 00:47:19.050
That is, it's possible to see
this happening as you're doing
00:47:19.050 --> 00:47:21.680
it if you're very careful.
00:47:21.680 --> 00:47:23.610
There's a mistake
in this process,
00:47:23.610 --> 00:47:26.290
and the mistake is
in the transition.
00:47:26.290 --> 00:47:28.230
This is a mistake here.
00:47:28.230 --> 00:47:33.340
Maybe I haven't used
any red yet today,
00:47:33.340 --> 00:47:34.940
so I get to use some red here.
00:47:34.940 --> 00:47:36.750
Oh boy.
00:47:36.750 --> 00:47:38.310
This is not true, here.
00:47:38.310 --> 00:47:39.200
This step here.
00:47:39.200 --> 00:47:40.470
So why isn't it true?
00:47:40.470 --> 00:47:43.500
It's not true for
the standard reason.
00:47:43.500 --> 00:47:50.670
Which is that really, x = plus
or minus square root of u.
00:47:50.670 --> 00:47:53.930
And if you stick to
one side or the other,
00:47:53.930 --> 00:47:55.660
you'll have a coherent
formula for it.
00:47:55.660 --> 00:47:58.243
One of them will be the plus and
one of them will be the minus
00:47:58.243 --> 00:48:01.980
and it will work out when you
separate it into its pieces.
00:48:01.980 --> 00:48:02.880
So you could do that.
00:48:02.880 --> 00:48:04.100
But this is a can of worms.
00:48:04.100 --> 00:48:06.250
So I avoid this.
00:48:06.250 --> 00:48:10.112
And just do it in a place where
the inverse is well defined.
00:48:10.112 --> 00:48:11.570
And where the
function either moves
00:48:11.570 --> 00:48:13.880
steadily up or steadily down.