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PROFESSOR: Hi.
00:00:33.380 --> 00:00:36.550
Having already studied area
and volume and its
00:00:36.550 --> 00:00:39.950
relationship to calculus, today,
we turn our attention
00:00:39.950 --> 00:00:41.330
to the study of length.
00:00:41.330 --> 00:00:44.860
And this may seem a bit strange
because, intuitively,
00:00:44.860 --> 00:00:48.170
I think it's fair to assume that
you would imagine that
00:00:48.170 --> 00:00:51.170
length would be simpler than
area which, in turn, would be
00:00:51.170 --> 00:00:54.050
simpler than volume and, hence,
that perhaps we should
00:00:54.050 --> 00:00:56.210
have started with length
in the first place.
00:00:56.210 --> 00:00:59.060
The interesting thing is in
terms of our structure, which
00:00:59.060 --> 00:01:01.920
we so far have called
two-dimensional area,
00:01:01.920 --> 00:01:05.099
three-dimensional area, and
which, today, we shall call
00:01:05.099 --> 00:01:09.220
one-dimensional area, a rather
peculiar thing that causes a
00:01:09.220 --> 00:01:12.470
great deal of difficulty,
intellectually speaking,
00:01:12.470 --> 00:01:16.850
occurs in the study of arc
length that does not occur in
00:01:16.850 --> 00:01:19.990
either the study of
area or volume.
00:01:19.990 --> 00:01:22.760
And I think that we'll start
our investigation today
00:01:22.760 --> 00:01:25.040
leading up to what this
really means.
00:01:25.040 --> 00:01:29.190
So, as I say, I call today's
lesson 'One-dimensional Area',
00:01:29.190 --> 00:01:30.400
which is arc length.
00:01:30.400 --> 00:01:34.660
And let's show that there is a
parallel, at least in part,
00:01:34.660 --> 00:01:37.780
between the structure of arc
length and the structure of
00:01:37.780 --> 00:01:39.140
area and volume.
00:01:39.140 --> 00:01:43.240
You may recall that for area,
our initial axiom was that the
00:01:43.240 --> 00:01:45.990
building block of area
was a rectangle.
00:01:45.990 --> 00:01:49.490
And for volumes, the building
block we saw was a cylinder.
00:01:49.490 --> 00:01:50.950
For arc length--
00:01:50.950 --> 00:01:52.560
I think it's fairly obvious
to guess what
00:01:52.560 --> 00:01:53.500
we're going to say--
00:01:53.500 --> 00:01:56.650
the basic building block is
a straight line segment.
00:01:56.650 --> 00:01:59.740
And so without further ado, that
becomes our first rule,
00:01:59.740 --> 00:02:02.230
our first axiom, axiom
number one.
00:02:02.230 --> 00:02:05.870
We assume that we can measure
the length of any straight
00:02:05.870 --> 00:02:07.440
line segment.
00:02:07.440 --> 00:02:09.050
That's our building block.
00:02:09.050 --> 00:02:12.560
The second axiom that we assume
is that the length of
00:02:12.560 --> 00:02:15.140
the whole equals the sum of
the lengths of the parts.
00:02:15.140 --> 00:02:18.290
In other words, if an arc is
broken down into constituent
00:02:18.290 --> 00:02:22.080
bases, the total arc length is
equal to the sum of the arc
00:02:22.080 --> 00:02:23.890
lengths of the constituent
parts.
00:02:23.890 --> 00:02:26.670
And at this stage, we can
say, so far, so good.
00:02:26.670 --> 00:02:28.500
This still looks like
it's going to be the
00:02:28.500 --> 00:02:30.390
same as area or volume.
00:02:30.390 --> 00:02:34.400
But now remember what one of the
axioms for both area and
00:02:34.400 --> 00:02:36.080
volume were, namely, what?
00:02:36.080 --> 00:02:40.120
That if region 'R' was contained
in region 'S', the
00:02:40.120 --> 00:02:45.530
area or the volume of 'R' was
no greater than that of the
00:02:45.530 --> 00:02:47.410
area or volume of 'S'.
00:02:47.410 --> 00:02:50.120
However, for arc length,
this is not true.
00:02:50.120 --> 00:02:51.330
It need not be true.
00:02:51.330 --> 00:02:52.410
I shouldn't say it's not true.
00:02:52.410 --> 00:02:56.580
It need not be true that if
region 'R' is contained in 'S'
00:02:56.580 --> 00:03:00.680
that the perimeter of region 'R'
is less than or equal to
00:03:00.680 --> 00:03:01.880
the perimeter of 'S'.
00:03:01.880 --> 00:03:05.850
In fact, this little diagram
that I've drawn over here, I
00:03:05.850 --> 00:03:08.680
hope will show you what
I'm driving at.
00:03:08.680 --> 00:03:11.290
Notice that it's rather clear
that the region 'R' here,
00:03:11.290 --> 00:03:15.080
which is shaded, is contained
inside the region 'S', which
00:03:15.080 --> 00:03:16.370
is my rectangle.
00:03:16.370 --> 00:03:18.930
And yet, if you look at the
perimeter here, all these
00:03:18.930 --> 00:03:22.620
finger-shaped things in here, I
think it's easy to see that
00:03:22.620 --> 00:03:27.600
the perimeter of 'R' exceeds
the perimeter of 'S'.
00:03:27.600 --> 00:03:30.510
And if it's not that easy to
see, heck, just make a few
00:03:30.510 --> 00:03:34.180
more loops inside here and
keep wiggling this thing
00:03:34.180 --> 00:03:37.370
around until you're convinced
that you have created this
00:03:37.370 --> 00:03:38.450
particular situation.
00:03:38.450 --> 00:03:42.790
All I want you to see here is
that it's plausible to you
00:03:42.790 --> 00:03:48.120
that we cannot talk about
lengths by squeezing them, as
00:03:48.120 --> 00:03:51.620
we did areas and volumes,
between regions that we
00:03:51.620 --> 00:03:54.800
already knew contained the
given region and were
00:03:54.800 --> 00:03:56.250
contained in the given region.
00:03:56.250 --> 00:03:58.910
Now let me just pause here for
one moment to make sure that
00:03:58.910 --> 00:04:00.140
we keep one thing straight.
00:04:00.140 --> 00:04:04.080
We're talking now about an
analytical approach to length.
00:04:04.080 --> 00:04:06.390
In other words, an approach that
will allow us to bring to
00:04:06.390 --> 00:04:10.980
bear all of the power of
calculus to the study.
00:04:10.980 --> 00:04:14.380
I don't want you to forget for
a moment that intuitively, we
00:04:14.380 --> 00:04:17.130
certainly do know what arc
length is, just as we
00:04:17.130 --> 00:04:21.100
intuitively had a feeling for
what area and volume were.
00:04:21.100 --> 00:04:23.780
Just to freshen our memories
on this, remember the
00:04:23.780 --> 00:04:24.960
intuitive approach.
00:04:24.960 --> 00:04:30.970
That if you have an arc from 'A'
to 'B', the typical way of
00:04:30.970 --> 00:04:33.910
measuring the arc length is to
take, for example, a piece of
00:04:33.910 --> 00:04:37.480
string, lay it off along the
curve from 'A' to 'B'.
00:04:37.480 --> 00:04:40.430
After you've done this,
pick the string up.
00:04:40.430 --> 00:04:43.610
And then straighten the string
out, whatever that means, and
00:04:43.610 --> 00:04:45.420
measure its length
with a ruler.
00:04:45.420 --> 00:04:48.020
And we won't worry about how
you know whether you're
00:04:48.020 --> 00:04:50.490
stretching the string too
taut or what have you.
00:04:50.490 --> 00:04:52.890
We'll leave out these
philosophic questions.
00:04:52.890 --> 00:04:56.500
All we'll say is we would like
a more objective method that
00:04:56.500 --> 00:04:59.250
will allow us to use
mathematical analysis.
00:04:59.250 --> 00:05:02.420
And so what we're going to try
to do next is to find an
00:05:02.420 --> 00:05:07.190
analytic way that will allow us
to use calculus, but at the
00:05:07.190 --> 00:05:11.150
same time will give us a
definition which agrees with
00:05:11.150 --> 00:05:12.360
our intuition.
00:05:12.360 --> 00:05:14.930
And the first question is
how shall we begin.
00:05:14.930 --> 00:05:18.890
And as so often is the case in
mathematics, we begin our new
00:05:18.890 --> 00:05:23.470
quest by going back to an
old way that worked
00:05:23.470 --> 00:05:24.750
for a previous case.
00:05:24.750 --> 00:05:27.640
And hopefully, we'll find a
way of extending the old
00:05:27.640 --> 00:05:29.640
situation to cover the new.
00:05:29.640 --> 00:05:33.280
Now what does this mean in
this particular instance?
00:05:33.280 --> 00:05:35.540
Well, let me just
call it this.
00:05:35.540 --> 00:05:38.110
I'll call it analytical
approach, trial number one.
00:05:38.110 --> 00:05:42.320
What I'm going to do is try to
imitate exactly what we did in
00:05:42.320 --> 00:05:43.620
the area case.
00:05:43.620 --> 00:05:47.290
For example, if I take the
region 'R', which I'll draw
00:05:47.290 --> 00:05:51.160
this way here, if this is the
region 'R', namely bounded
00:05:51.160 --> 00:05:55.430
above by the curve 'y' equals 'f
of x', below by the x-axis,
00:05:55.430 --> 00:05:58.000
on the left, by the line 'x'
equals 'a', and on the right,
00:05:58.000 --> 00:06:01.010
by the line 'x' equals 'b', how
did we find the area of
00:06:01.010 --> 00:06:01.860
the region 'R'?
00:06:01.860 --> 00:06:06.020
Well, what we did is we
inscribed and we circumscribed
00:06:06.020 --> 00:06:07.150
rectangles.
00:06:07.150 --> 00:06:10.132
And we took the limit of the
circumscribed rectangles, et
00:06:10.132 --> 00:06:13.270
cetera, and put the squeeze on
as 'n' went to infinity.
00:06:13.270 --> 00:06:16.990
Now the idea is we might get the
idea that maybe we should
00:06:16.990 --> 00:06:18.470
do the same thing
for arc length.
00:06:18.470 --> 00:06:21.770
In other words, let me call one
of these little pieces of
00:06:21.770 --> 00:06:23.540
arc length 'delta w'.
00:06:23.540 --> 00:06:24.990
In other words, I'm just
isolating part
00:06:24.990 --> 00:06:26.370
of the diagram here.
00:06:26.370 --> 00:06:27.640
Here's 'delta w'.
00:06:27.640 --> 00:06:28.770
Here's 'delta x'.
00:06:28.770 --> 00:06:30.420
Here's 'delta y'.
00:06:30.420 --> 00:06:34.810
The idea is in the same way that
I approximated a piece of
00:06:34.810 --> 00:06:37.830
area by an inscribed and a
circumscribed rectangle, why
00:06:37.830 --> 00:06:41.660
can't I say something like,
well, let me let 'delta w' be
00:06:41.660 --> 00:06:44.410
approximately equal
to 'delta x'?
00:06:44.410 --> 00:06:46.820
And just to make sure that our
memories are refreshed over
00:06:46.820 --> 00:06:51.950
here, notice that 'delta x' is
just the length of each piece
00:06:51.950 --> 00:06:55.860
if the segment from 'a' to 'b',
namely of length 'b - a',
00:06:55.860 --> 00:06:58.670
is divided into 'n'
equal parts.
00:06:58.670 --> 00:07:01.710
See, the idea is why can't we
mimic the same approach.
00:07:01.710 --> 00:07:05.140
And let me point out what is
so crucial here in terms of
00:07:05.140 --> 00:07:08.310
what I mentioned above, namely,
notice that the reason
00:07:08.310 --> 00:07:11.710
that we can say that the area of
the region 'R' is just the
00:07:11.710 --> 00:07:14.920
limit of 'U sub n' as 'n'
approaches infinity, where 'U
00:07:14.920 --> 00:07:18.000
sub n' is the area of the
circumscribed rectangles.
00:07:18.000 --> 00:07:21.950
The only reason we can say that
is because we squeezed 'A
00:07:21.950 --> 00:07:27.540
sub r' between 'L sub n', the
inscribed rectangles, and 'U
00:07:27.540 --> 00:07:29.990
sub n', the circumscribed
rectangles.
00:07:29.990 --> 00:07:32.310
And the limits of these
lower bounds and
00:07:32.310 --> 00:07:34.030
upper bounds were equal.
00:07:34.030 --> 00:07:36.830
'A sub 'r was squeezed
between these two.
00:07:36.830 --> 00:07:39.170
Hence, it had to equal
the common limit.
00:07:39.170 --> 00:07:41.370
That was the structure
that we used.
00:07:41.370 --> 00:07:44.910
On the other hand, we can't use
that when we're dealing
00:07:44.910 --> 00:07:46.060
with arc length.
00:07:46.060 --> 00:07:48.450
And I'll mention that in
a few moments again.
00:07:48.450 --> 00:07:50.860
But let me just point out what
I'm driving at this way.
00:07:50.860 --> 00:07:53.380
Suppose we mimic this
as we did before.
00:07:53.380 --> 00:07:56.620
And we say, OK, let the element
of arc length, 'delta
00:07:56.620 --> 00:07:59.720
w', be approximately
equal to 'delta x'.
00:07:59.720 --> 00:08:02.410
And now what I will do
is define script
00:08:02.410 --> 00:08:04.500
'L' from 'a' to 'b'.
00:08:04.500 --> 00:08:06.570
I don't want to call it arc
length because it may not be.
00:08:06.570 --> 00:08:10.090
But as a first approximation,
let me define this symbol to
00:08:10.090 --> 00:08:14.640
be the limit of the sum of all
these 'delta x's when we
00:08:14.640 --> 00:08:18.270
divide this region into 'n'
parts as 'n' goes to infinity.
00:08:18.270 --> 00:08:21.060
Now look, I have the right to
make up this particular
00:08:21.060 --> 00:08:22.540
definition.
00:08:22.540 --> 00:08:25.080
Now if I compute this
limit, what happens?
00:08:25.080 --> 00:08:28.280
Recall that we mentioned
that 'delta x' was
00:08:28.280 --> 00:08:30.060
'b - a' over 'n'.
00:08:30.060 --> 00:08:33.620
Consequently, if I have 'n' of
these pieces, the total sum
00:08:33.620 --> 00:08:34.490
would be what?
00:08:34.490 --> 00:08:37.380
'n' times b' - a' over 'n'.
00:08:37.380 --> 00:08:40.995
And 'n' times 'b - a' over
'n' is just 'b - a'.
00:08:40.995 --> 00:08:45.170
In other words, script 'L' from
'a' to 'b' is defined.
00:08:45.170 --> 00:08:47.880
And it's 'b - a', not 'w'.
00:08:47.880 --> 00:08:49.550
In other words, coming
back to our
00:08:49.550 --> 00:08:51.460
diagram, notice what happened.
00:08:51.460 --> 00:08:54.370
What we wanted was a recipe
that would give
00:08:54.370 --> 00:08:56.160
us this length here.
00:08:56.160 --> 00:09:00.600
What we found was a recipe that
gave us the length from
00:09:00.600 --> 00:09:01.730
'a' to 'b'.
00:09:01.730 --> 00:09:05.610
Now intuitively, we know that
the length from 'a' to 'b' is
00:09:05.610 --> 00:09:07.620
not the arc length that
we're looking for.
00:09:07.620 --> 00:09:10.520
In other words, what we defined
to be script 'L'
00:09:10.520 --> 00:09:14.290
existed as a limit, but it gave
us an answer which did
00:09:14.290 --> 00:09:16.530
not coincide with
our intuition.
00:09:16.530 --> 00:09:19.480
Since we intuitively know what
the right answer is, we must
00:09:19.480 --> 00:09:22.250
discard this approach in the
sense that it doesn't give us
00:09:22.250 --> 00:09:24.330
an answer that we have
any faith in.
00:09:24.330 --> 00:09:27.550
And by the way, notice where we
went wrong over here if you
00:09:27.550 --> 00:09:29.200
want to look at it from
that point of view.
00:09:29.200 --> 00:09:31.850
Notice that when we approximated
'delta w' by
00:09:31.850 --> 00:09:36.450
'delta x', it's clear from this
diagram that 'delta x'
00:09:36.450 --> 00:09:38.370
was certainly less
than 'delta w'.
00:09:38.370 --> 00:09:41.570
But notice that we didn't have
an upper bound here.
00:09:41.570 --> 00:09:45.270
Or we can make speculations
like, maybe 'delta x + delta
00:09:45.270 --> 00:09:48.320
y' would be more than 'delta
w', and things like this.
00:09:48.320 --> 00:09:50.310
We'll talk about that
more later.
00:09:50.310 --> 00:09:53.720
But for now, all I want us
to see is the degree of
00:09:53.720 --> 00:09:57.420
sophistication that enters into
the arc length problem
00:09:57.420 --> 00:10:00.160
that didn't bother us in either
the area or the volume
00:10:00.160 --> 00:10:04.670
problems, namely, we are missing
now the all-important
00:10:04.670 --> 00:10:06.430
squeeze element.
00:10:06.430 --> 00:10:09.050
Well, no sense crying
over spilt milk.
00:10:09.050 --> 00:10:12.560
We go on, and we try the
next type of approach.
00:10:12.560 --> 00:10:16.510
In other words, what we sense
now is why don't we do this.
00:10:16.510 --> 00:10:20.730
Instead of approximating 'delta
w' by 'delta x', why
00:10:20.730 --> 00:10:24.730
don't we approximate 'delta w'
by the cord that joins the two
00:10:24.730 --> 00:10:27.070
endpoints of the arc.
00:10:27.070 --> 00:10:29.070
In other words, I think that
we began to suspect
00:10:29.070 --> 00:10:32.530
intuitively that, somehow or
other, for a small change in
00:10:32.530 --> 00:10:36.200
'delta x', 'delta s' should be
a better approximation to
00:10:36.200 --> 00:10:38.660
'delta w' than 'delta x' was.
00:10:38.660 --> 00:10:41.720
Of course, the wide open
question is granted that it's
00:10:41.720 --> 00:10:43.710
better, is it good enough.
00:10:43.710 --> 00:10:46.360
Well, we'll worry about that in
a little more detail later.
00:10:46.360 --> 00:10:49.670
All we're saying is let 'delta
w' be approximately
00:10:49.670 --> 00:10:50.620
equal to 'delta s'.
00:10:50.620 --> 00:10:52.330
In other words, we'll
approximate 'delta
00:10:52.330 --> 00:10:54.100
w' by 'delta s'.
00:10:54.100 --> 00:10:58.230
And we'll now define 'L' from
'a' to 'b', 'L' from 'a' to
00:10:58.230 --> 00:11:02.160
'b' to be the limit not of the
sum of 'delta x's now, but the
00:11:02.160 --> 00:11:06.500
sum of the 'delta s's, as 'k'
goes from 1 to 'n', taken in
00:11:06.500 --> 00:11:08.790
the limit as 'n' goes
to infinity.
00:11:08.790 --> 00:11:11.480
And for those of us who are more
familiar with 'delta x's
00:11:11.480 --> 00:11:15.060
and 'delta y's, and the symbol
delta s bothers us, simply
00:11:15.060 --> 00:11:18.750
observe that by the Pythagorean
theorem, 'delta s'
00:11:18.750 --> 00:11:22.000
is related to 'delta x' and
'delta y' by ''delta s'
00:11:22.000 --> 00:11:23.860
squared' equals ''delta
x' squared'
00:11:23.860 --> 00:11:25.290
plus ''delta y' squared'.
00:11:25.290 --> 00:11:28.490
So we can rewrite this in
this particular form.
00:11:28.490 --> 00:11:32.570
In other words, I will define
capital 'L' hopefully to stand
00:11:32.570 --> 00:11:33.590
for length later on.
00:11:33.590 --> 00:11:35.490
But we'll worry about
that later too.
00:11:35.490 --> 00:11:39.370
But 'L' from 'a' to 'b' to
be this particular limit.
00:11:39.370 --> 00:11:43.200
And now I claim that there are
three natural questions with
00:11:43.200 --> 00:11:45.370
which we must come to grips.
00:11:45.370 --> 00:11:48.830
The first question is does
this limit even exist.
00:11:48.830 --> 00:11:50.500
Does this limit exist?
00:11:50.500 --> 00:11:51.920
And the answer is
that, except for
00:11:51.920 --> 00:11:54.200
far-fetched curves, it does.
00:11:54.200 --> 00:12:00.040
You really have to get a curve
that wiggles uncontrollably to
00:12:00.040 --> 00:12:02.930
break the possibility of
this limit existing.
00:12:02.930 --> 00:12:06.520
Unfortunately, there are
pathological cases, one of
00:12:06.520 --> 00:12:10.120
which is described in the text
assignment for this lesson, of
00:12:10.120 --> 00:12:13.110
a curve that doesn't have a
finite limit when you try to
00:12:13.110 --> 00:12:14.730
compute the arc length
this way.
00:12:14.730 --> 00:12:15.920
Just a little idiosyncrasy.
00:12:15.920 --> 00:12:19.110
However, for any curve that
comes up in real life, that
00:12:19.110 --> 00:12:22.250
doesn't oscillate too violently
with infinite
00:12:22.250 --> 00:12:25.210
variations, et cetera, et
cetera, which we won't, again,
00:12:25.210 --> 00:12:28.920
talk about right now, the idea
is that this limit does exist.
00:12:28.920 --> 00:12:31.990
As far as this course is
concerned, we shall assume the
00:12:31.990 --> 00:12:33.940
answer to question one is yes.
00:12:33.940 --> 00:12:36.520
In fact, the way we'll do it
without being dictatorial is
00:12:36.520 --> 00:12:38.795
we'll say, look, if this limit
doesn't exist, we just won't
00:12:38.795 --> 00:12:40.120
study that curve.
00:12:40.120 --> 00:12:42.840
In fact, we will call
a curve rectifiable
00:12:42.840 --> 00:12:44.510
if this limit exists.
00:12:44.510 --> 00:12:47.460
And so we'll assume that we
deal only with rectifiable
00:12:47.460 --> 00:12:50.040
curves, in other words, that
this limit does exist.
00:12:50.040 --> 00:12:51.380
Question number two.
00:12:51.380 --> 00:12:52.970
OK, the limit exists.
00:12:52.970 --> 00:12:54.790
So how do we compute it?
00:12:54.790 --> 00:12:58.060
And that, in general, is not a
very easy thing to answer.
00:12:58.060 --> 00:13:00.830
What's even worse though is
that after you've answered
00:13:00.830 --> 00:13:03.870
this, you have to come to grips
with a question that we
00:13:03.870 --> 00:13:07.480
were able to dodge when we
studied both area and volume,
00:13:07.480 --> 00:13:11.370
namely, the question is once
this limit does exist and you
00:13:11.370 --> 00:13:14.750
compute it, how do you know
that it agrees with our
00:13:14.750 --> 00:13:16.900
intuitive definition
of arc length.
00:13:16.900 --> 00:13:19.230
In other words, if you recall
what we did just a few minutes
00:13:19.230 --> 00:13:23.330
ago, we defined script 'L'
from 'a' to 'b' to be a
00:13:23.330 --> 00:13:24.290
certain limit.
00:13:24.290 --> 00:13:26.800
We showed that that
limit existed.
00:13:26.800 --> 00:13:30.560
The problem was is that limit,
even though it existed, did
00:13:30.560 --> 00:13:34.350
not give us an answer that
agreed intuitively with what
00:13:34.350 --> 00:13:36.700
we believed arc length
was supposed to mean.
00:13:36.700 --> 00:13:38.810
In other words, you see, we've
assumed the answer to the
00:13:38.810 --> 00:13:40.820
first question is yes.
00:13:40.820 --> 00:13:42.640
Now we have two questions
to answer.
00:13:42.640 --> 00:13:45.290
How do you compute this limit,
which is a hard question in
00:13:45.290 --> 00:13:46.080
it's own right?
00:13:46.080 --> 00:13:49.110
Secondly, once you do compute
this limit, how do you know
00:13:49.110 --> 00:13:52.340
that it's going to agree with
the intuitive answer that you
00:13:52.340 --> 00:13:53.430
get for arc length?
00:13:53.430 --> 00:13:56.510
And this shall be what we have
to answer in the remainder of
00:13:56.510 --> 00:13:58.140
our lesson today.
00:13:58.140 --> 00:13:59.680
Let's take these in order.
00:13:59.680 --> 00:14:02.370
And let's try to answer question
number two first.
00:14:02.370 --> 00:14:05.470
The idea is we've defined
capital 'L' from 'a' to 'b' to
00:14:05.470 --> 00:14:07.770
be this particular limit, and
we'd like to know if this
00:14:07.770 --> 00:14:09.000
limit exists.
00:14:09.000 --> 00:14:12.370
Not only that, but we have a
great command of calculus at
00:14:12.370 --> 00:14:13.590
our disposal now.
00:14:13.590 --> 00:14:17.490
All of the previous lessons can
be brought to bear here to
00:14:17.490 --> 00:14:20.470
help us put this into the
perspective of what calculus
00:14:20.470 --> 00:14:21.380
is all about.
00:14:21.380 --> 00:14:24.890
For example, when I see an
expression like this, I like
00:14:24.890 --> 00:14:26.670
to think in terms
of a derivative.
00:14:26.670 --> 00:14:29.660
A derivative reminds me of
'delta y' divided by 'delta
00:14:29.660 --> 00:14:30.810
x', et cetera.
00:14:30.810 --> 00:14:33.910
So what I do here is I factor
out a ''delta x' squared'.
00:14:33.910 --> 00:14:36.750
In other words, I divide through
by ''delta x' squared'
00:14:36.750 --> 00:14:39.390
inside the radical sign, which
is really the same
00:14:39.390 --> 00:14:41.920
equivalently as dividing
by 'delta x'.
00:14:41.920 --> 00:14:44.230
And I multiply by 'delta
x' outside.
00:14:44.230 --> 00:14:46.840
In other words, factoring out
with ''delta x' squared', the
00:14:46.840 --> 00:14:49.100
square root of ''delta x'
squared' plus ''delta y'
00:14:49.100 --> 00:14:53.060
squared' can be written as the
square root of '1 + ''delta y'
00:14:53.060 --> 00:14:56.240
over 'delta x'' squared'
times 'delta x'.
00:14:56.240 --> 00:14:59.900
Now the idea is that 'delta y'
over 'delta x' is the slope of
00:14:59.900 --> 00:15:03.910
that cord that joins the two
endpoints of 'delta w'.
00:15:03.910 --> 00:15:05.490
It's not a derivative
as we know it.
00:15:05.490 --> 00:15:07.750
It's the slope of a straight
line cord, not
00:15:07.750 --> 00:15:09.220
the slope of a curve.
00:15:09.220 --> 00:15:10.750
Now the whole idea is this.
00:15:10.750 --> 00:15:14.040
We know from the mean value
theorem that if our curve is
00:15:14.040 --> 00:15:17.820
smooth, there is a point in
the interval at which the
00:15:17.820 --> 00:15:20.690
derivative at that point
is equal to the
00:15:20.690 --> 00:15:22.020
slope of the cord.
00:15:22.020 --> 00:15:25.170
In other words, if 'f' is
differentiable on [a, b], we
00:15:25.170 --> 00:15:27.170
may invoke the Mean
Value Theorem--
00:15:27.170 --> 00:15:30.280
here abbreviated as MVT, the
Mean Value Theorem--
00:15:30.280 --> 00:15:34.660
to conclude that there is some
point 'c sub k' in our 'delta
00:15:34.660 --> 00:15:38.680
x' interval for which ''delta y'
over 'delta x'' is 'f prime
00:15:38.680 --> 00:15:39.860
of 'c sub k''.
00:15:39.860 --> 00:15:42.420
And in order to help you
facilitate what we're talking
00:15:42.420 --> 00:15:45.005
about in your minds, look at
the following diagram.
00:15:45.005 --> 00:15:46.110
This is all we're saying.
00:15:46.110 --> 00:15:48.350
What we're saying is here's
our 'delta x',
00:15:48.350 --> 00:15:49.520
here's our 'delta y'.
00:15:49.520 --> 00:15:52.870
We'll call this point 'x
sub 'k - 1'', this
00:15:52.870 --> 00:15:54.240
point 'x sub k'.
00:15:54.240 --> 00:15:56.020
This is our k-th partition.
00:15:56.020 --> 00:15:59.270
'Delta y' divided by 'delta
x' is just the
00:15:59.270 --> 00:16:00.850
slope of this line.
00:16:00.850 --> 00:16:03.630
See, that's just the
slope of this line.
00:16:03.630 --> 00:16:05.825
And what the Mean Value Theorem
says is if this curve
00:16:05.825 --> 00:16:10.280
is smooth, some place on this
arc, there is a point where
00:16:10.280 --> 00:16:16.380
the line tangent to the curve
is parallel to this cord.
00:16:16.380 --> 00:16:18.980
And that's what I'm calling
the point 'c sub k'.
00:16:18.980 --> 00:16:23.170
'c sub k' is the point at which
the slope of the curve
00:16:23.170 --> 00:16:25.430
is equal to the slope
of the cord.
00:16:25.430 --> 00:16:29.290
In other words, if 'f' is
continuous, I can conclude
00:16:29.290 --> 00:16:34.880
that 'L' from 'a' to 'b' is the
limit as 'n' approaches
00:16:34.880 --> 00:16:39.500
infinity, summation 'k' goes
from 1 to 'n', square root of
00:16:39.500 --> 00:16:44.830
'1 + ''f prime 'c sub k''
squared' times 'delta x'.
00:16:44.830 --> 00:16:49.650
And notice that this now
starts to look like my
00:16:49.650 --> 00:16:52.530
definite integral according to
the definition that we were
00:16:52.530 --> 00:16:55.720
talking about in our earlier
lectures in this block.
00:16:55.720 --> 00:16:59.070
In fact, how can we invoke the
first fundamental theorem of
00:16:59.070 --> 00:17:00.340
integral calculus?
00:17:00.340 --> 00:17:04.583
Remember, if this expression
here-- it's
00:17:04.583 --> 00:17:05.829
not an integral yet--
00:17:05.829 --> 00:17:08.589
happens to be a continuous
function, then we're in pretty
00:17:08.589 --> 00:17:09.700
good shape.
00:17:09.700 --> 00:17:12.630
In other words, if I can assume
that 'f prime' is
00:17:12.630 --> 00:17:13.240
continuous--
00:17:13.240 --> 00:17:15.470
let's go over here and
continue on here.
00:17:15.470 --> 00:17:20.660
See, what I'm saying is if I can
assume that 'f prime' is
00:17:20.660 --> 00:17:24.200
continuous, well, look, the
square of a continuous
00:17:24.200 --> 00:17:25.980
function is continuous.
00:17:25.980 --> 00:17:29.280
The sum of two continuous
functions is continuous.
00:17:29.280 --> 00:17:31.900
And the square root of a
continuous function is
00:17:31.900 --> 00:17:32.830
continuous.
00:17:32.830 --> 00:17:36.150
In other words, and this is a
key point, if the derivative
00:17:36.150 --> 00:17:41.380
is continuous, I can conclude
that the 'L' from 'a' to 'b'
00:17:41.380 --> 00:17:43.650
can be replaced by the definite
integral from 'a' to
00:17:43.650 --> 00:17:48.690
'b' square root of '1 + ''dy/dx'
squared'' times 'dx',
00:17:48.690 --> 00:17:54.150
which I quickly point out
may be hard to evaluate.
00:17:54.150 --> 00:17:57.370
In other words, one thing I
could try to do over here is
00:17:57.370 --> 00:18:01.110
to find the function g whose
derivative with respect to 'x'
00:18:01.110 --> 00:18:04.000
is the square root of '1 +
''dy/dx' squared'' and
00:18:04.000 --> 00:18:05.890
evaluate that between
'a' and 'b'.
00:18:05.890 --> 00:18:10.850
I can put approximations on
here, whatever I want.
00:18:10.850 --> 00:18:12.500
In fact, let's summarize
it down here.
00:18:12.500 --> 00:18:16.220
If 'f' is differentiable on the
closed interval from 'a'
00:18:16.220 --> 00:18:20.000
to 'b' and if 'f prime'
is the derivative--
00:18:20.000 --> 00:18:21.320
you see, 'f prime'--
00:18:21.320 --> 00:18:23.750
is also continuous on the closed
interval from 'a' to
00:18:23.750 --> 00:18:27.250
'b', then not only does capital
'L' from 'a' to 'b'
00:18:27.250 --> 00:18:30.410
exist, but it's given
computationally by this
00:18:30.410 --> 00:18:32.150
particular integral.
00:18:32.150 --> 00:18:35.220
And that answers question number
two, that the limit
00:18:35.220 --> 00:18:38.000
exists, and this is what
it's equal to.
00:18:38.000 --> 00:18:40.110
The problem that we're
faced with--
00:18:40.110 --> 00:18:41.180
and I've written this out.
00:18:41.180 --> 00:18:44.010
I think it looks harder
than what it says.
00:18:44.010 --> 00:18:46.825
But I've taken the trouble to
write this whole thing out, so
00:18:46.825 --> 00:18:49.870
that if you have trouble
following what I'm saying,
00:18:49.870 --> 00:18:52.530
that you can see this thing
blocked out for you.
00:18:52.530 --> 00:18:53.590
The idea is this.
00:18:53.590 --> 00:18:57.060
What we have done is we have
approximated 'delta
00:18:57.060 --> 00:18:59.170
w' by 'delta s'.
00:18:59.170 --> 00:19:02.390
Then what we said is 'w'
is the sum of all
00:19:02.390 --> 00:19:03.900
these 'delta w's.
00:19:03.900 --> 00:19:08.460
And since each 'delta w' is
approximately 'delta s', then
00:19:08.460 --> 00:19:12.340
what we can be sure of is that
'w' is approximated by this
00:19:12.340 --> 00:19:13.730
sum over here.
00:19:13.730 --> 00:19:14.740
Now here's what we did.
00:19:14.740 --> 00:19:17.050
We didn't work with 'w'
at all after this.
00:19:17.050 --> 00:19:20.000
We turned our attention
to this.
00:19:20.000 --> 00:19:21.440
This is what we did
in our case here.
00:19:21.440 --> 00:19:24.750
And we showed that this
limit existed.
00:19:24.750 --> 00:19:29.810
We showed that the limit, as
'k' went from 1 to 'n' and
00:19:29.810 --> 00:19:33.790
then went to infinity of these
pieces here, was 'L' of 'ab'.
00:19:33.790 --> 00:19:34.770
And that existed.
00:19:34.770 --> 00:19:39.600
What we did not show is that
this limit was w itself.
00:19:39.600 --> 00:19:42.940
Intuitively, you might say, if
I put the squeeze on this,
00:19:42.940 --> 00:19:45.670
doesn't this get rid of
all the error for me?
00:19:45.670 --> 00:19:48.930
We haven't shown that we've
gotten rid of all the error.
00:19:48.930 --> 00:19:51.750
In essence, how do we know
if all the error has been
00:19:51.750 --> 00:19:53.000
squeezed out?
00:19:53.000 --> 00:19:56.210
This is precisely what question
three is all about.
00:19:56.210 --> 00:19:59.530
Again, going back to what we did
earlier, remember, when we
00:19:59.530 --> 00:20:03.330
approximated 'delta w' by 'delta
x', then we said, OK,
00:20:03.330 --> 00:20:06.380
add up all these 'delta x's, and
take the limit as 'n' goes
00:20:06.380 --> 00:20:07.190
to infinity.
00:20:07.190 --> 00:20:11.090
We found that that limit was
'b - a', which was not the
00:20:11.090 --> 00:20:12.670
length of the curve.
00:20:12.670 --> 00:20:14.750
In other words, somehow or
other, even though the limit
00:20:14.750 --> 00:20:17.760
existed, we did not squeeze
out all the error.
00:20:17.760 --> 00:20:21.220
And this is why the study of
arc length is so difficult.
00:20:21.220 --> 00:20:23.210
Because we don't have a
sandwiching effect.
00:20:23.210 --> 00:20:27.210
It is very difficult for us to
figure out when we've squeezed
00:20:27.210 --> 00:20:28.350
out all the error.
00:20:28.350 --> 00:20:30.290
So at any rate, let
me generalize
00:20:30.290 --> 00:20:32.880
question number three.
00:20:32.880 --> 00:20:34.120
Remember what question
number three is?
00:20:34.120 --> 00:20:36.510
How do we know that if
the limit exists,
00:20:36.510 --> 00:20:37.610
it's equal to 'w'?
00:20:37.610 --> 00:20:40.860
All I'm saying is don't even
worry about arc length.
00:20:40.860 --> 00:20:44.420
Just suppose that 'w' is any
function defined on a closed
00:20:44.420 --> 00:20:47.470
interval from 'a' to 'b' and
that we've approximated 'delta
00:20:47.470 --> 00:20:52.350
w' by something of the form 'g
of 'c sub k'' times 'delta x',
00:20:52.350 --> 00:20:55.700
where 'g' is what I call
some intuitive function
00:20:55.700 --> 00:20:57.140
defined on [a, b].
00:20:57.140 --> 00:21:01.160
For example, in our earlier
example, we started with
00:21:01.160 --> 00:21:02.820
'delta w' being arc length.
00:21:02.820 --> 00:21:06.460
And we approximated 'delta w' by
'delta x' in which case 'g'
00:21:06.460 --> 00:21:09.220
would've been the function
which is identically 1.
00:21:09.220 --> 00:21:12.130
In the area situation, remember
we approximated
00:21:12.130 --> 00:21:15.120
'delta A' by something
times 'delta x'.
00:21:15.120 --> 00:21:16.860
Well, what times 'delta x'?
00:21:16.860 --> 00:21:18.570
Well, it was the height
of a rectangle.
00:21:18.570 --> 00:21:21.370
In other words, we look at the
thing we're trying to find, we
00:21:21.370 --> 00:21:22.370
use our intuition--
00:21:22.370 --> 00:21:25.090
and this is difficult because
intuition varies from person
00:21:25.090 --> 00:21:25.890
to person--
00:21:25.890 --> 00:21:28.290
and we say, what would make
a good approximation here.
00:21:28.290 --> 00:21:30.600
What would be an
approximation?
00:21:30.600 --> 00:21:35.770
We say, OK, let's approximate
'delta w' by 'g of 'c sub k''
00:21:35.770 --> 00:21:38.420
times 'delta x', where 'c'
is some point in the
00:21:38.420 --> 00:21:39.570
interval, et cetera.
00:21:39.570 --> 00:21:42.290
Then we add up all of these
'delta w's as 'k'
00:21:42.290 --> 00:21:43.800
goes from 1 to 'n'.
00:21:43.800 --> 00:21:45.830
We say, OK, that's
approximately
00:21:45.830 --> 00:21:47.420
this thing over here.
00:21:47.420 --> 00:21:51.750
Now what we have shown is that
if 'g' is continuous on [a, b]
00:21:51.750 --> 00:21:57.200
then as 'n' goes to infinity,
this particular limit exists
00:21:57.200 --> 00:21:59.210
and is denoted by the
integral from 'a' to
00:21:59.210 --> 00:22:01.840
'b', ''g of x' dx'.
00:22:01.840 --> 00:22:03.900
This is what we've
shown so far.
00:22:03.900 --> 00:22:07.320
What the big question is is,
granted that this limit
00:22:07.320 --> 00:22:09.810
exists, does it equal 'w'?
00:22:09.810 --> 00:22:13.280
In other words, is 'w' equal
to the integral from 'a' to
00:22:13.280 --> 00:22:15.870
b', ''g of x' dx'?
00:22:15.870 --> 00:22:19.060
That's what the remainder of
today's lesson is about as far
00:22:19.060 --> 00:22:20.400
as arc length is concerned.
00:22:20.400 --> 00:22:23.170
And I'm going to solve this
problem in general first and
00:22:23.170 --> 00:22:25.420
then make some applications
about this
00:22:25.420 --> 00:22:27.590
to arc length itself.
00:22:27.590 --> 00:22:30.560
And by the way, what we're going
to see next is you may
00:22:30.560 --> 00:22:33.610
remember that very, very early
in our course, we came to
00:22:33.610 --> 00:22:36.220
grips with something called
infinitesimals.
00:22:36.220 --> 00:22:40.630
We came to grips with this delta
y tan infinitesimals of
00:22:40.630 --> 00:22:41.520
higher order.
00:22:41.520 --> 00:22:45.120
And now we're going to see how
just as this came up in
00:22:45.120 --> 00:22:49.170
differential calculus, these
same problems of approximation
00:22:49.170 --> 00:22:51.200
come up in integral calculus.
00:22:51.200 --> 00:22:54.020
The only difference, as we've
mentioned before, is instead
00:22:54.020 --> 00:22:56.690
of having to come to grips with
the indeterminate form
00:22:56.690 --> 00:22:59.890
0/0, we're going to have to
come to grips with the
00:22:59.890 --> 00:23:02.700
indeterminate form
infinity times 0.
00:23:02.700 --> 00:23:04.560
Let me show you what
I mean by that.
00:23:04.560 --> 00:23:05.550
The idea is this.
00:23:05.550 --> 00:23:08.920
Let's suppose that our case
'delta w'-- we've broken up
00:23:08.920 --> 00:23:10.800
'w' now into increments--
00:23:10.800 --> 00:23:13.960
and let's suppose that we're
approximating 'delta w', as we
00:23:13.960 --> 00:23:18.500
said before, by 'g of 'c sub
k'' times 'delta x'.
00:23:18.500 --> 00:23:20.800
Well, what do we mean by we're
approximating this?
00:23:20.800 --> 00:23:23.400
What we mean is there's
some error in here.
00:23:23.400 --> 00:23:27.110
Let's call the error 'alpha
sub k' times 'delta x'.
00:23:27.110 --> 00:23:29.960
In other words, this is just
a correction factor.
00:23:29.960 --> 00:23:32.260
This is what we have to
add on to this to make
00:23:32.260 --> 00:23:34.130
this equality whole.
00:23:34.130 --> 00:23:37.510
Once I add on the error, I'm
no longer working with an
00:23:37.510 --> 00:23:38.230
inequality.
00:23:38.230 --> 00:23:40.920
I'm working with an equality.
00:23:40.920 --> 00:23:43.130
And that allows me to
use some theorems.
00:23:43.130 --> 00:23:47.140
What I can say now is by
definition, w is the sum of
00:23:47.140 --> 00:23:48.720
all these 'delta w's.
00:23:48.720 --> 00:23:52.120
But 'delta w' being a sum, we
can use theorems about the
00:23:52.120 --> 00:23:53.390
sigma notation.
00:23:53.390 --> 00:23:56.080
In other words, what is the sum
of all these 'delta w's?
00:23:56.080 --> 00:23:59.200
It's the sum of all of these
pieces plus the sum of all of
00:23:59.200 --> 00:24:01.700
these pieces, which I've
written over here.
00:24:01.700 --> 00:24:05.130
And now you see, if I transpose,
I get that 'w'
00:24:05.130 --> 00:24:09.330
minus this sum is equal to the
'sum k' goes from 1 to 'n',
00:24:09.330 --> 00:24:12.260
'alpha k' times 'delta x'.
00:24:12.260 --> 00:24:14.670
Now the next thing I do
is take the limit
00:24:14.670 --> 00:24:16.600
as 'n' goes to infinity.
00:24:16.600 --> 00:24:20.810
By definition, since 'g' is a
continuous function, this
00:24:20.810 --> 00:24:23.350
limit here is just the definite
integral from 'a' to
00:24:23.350 --> 00:24:25.530
'b', ''g of x' dx'.
00:24:25.530 --> 00:24:28.490
On the other hand, this limit
here is what we have to
00:24:28.490 --> 00:24:29.560
investigate.
00:24:29.560 --> 00:24:32.500
In other words, we would like
to know whether 'w' is equal
00:24:32.500 --> 00:24:34.010
to the definite integral
or not.
00:24:34.010 --> 00:24:36.910
If we look at this particular
equation, what we have now
00:24:36.910 --> 00:24:40.350
shown is whatever the
relationship is between these
00:24:40.350 --> 00:24:44.130
two terms, it's typified by the
fact that this difference
00:24:44.130 --> 00:24:45.810
is this particular limit.
00:24:45.810 --> 00:24:48.840
In other words, if this limit
happens to be 0, then the
00:24:48.840 --> 00:24:52.330
integral will equal what we're
setting out to show it's equal
00:24:52.330 --> 00:24:53.850
to, namely, this function
itself.
00:24:53.850 --> 00:24:57.450
On the other hand, what we're
saying is we do not know that
00:24:57.450 --> 00:24:58.770
this limit is 0.
00:24:58.770 --> 00:25:00.470
By the way, notice what's
happening over here.
00:25:00.470 --> 00:25:04.020
As 'n' goes to infinity, 'delta
x' is going to 0.
00:25:04.020 --> 00:25:07.260
In other words, each individual
term in the sum is
00:25:07.260 --> 00:25:10.540
going to 0, but the number of
pieces is becoming infinite.
00:25:10.540 --> 00:25:13.340
There's your infinity
times 0 form here.
00:25:13.340 --> 00:25:16.420
And let me show you a case where
the pieces are growing
00:25:16.420 --> 00:25:20.170
too fast in number to be offset
by the fact that their
00:25:20.170 --> 00:25:21.730
size is going to 0.
00:25:21.730 --> 00:25:25.110
For the sake of argument, let me
suppose that 'alpha sub k'
00:25:25.110 --> 00:25:29.100
happens to be some non-0
constant for all 'k'.
00:25:29.100 --> 00:25:33.110
If I come back to this
expression here, if 'alpha sub
00:25:33.110 --> 00:25:36.310
k' is equal to a constant, I'll
replace 'alpha sub k' by
00:25:36.310 --> 00:25:38.300
that constant, which is 'c'.
00:25:38.300 --> 00:25:39.450
I now have what?
00:25:39.450 --> 00:25:42.340
That the limit that I'm looking
for is the 'sum k'
00:25:42.340 --> 00:25:46.030
goes from 1 to 'n', 'c' times
'delta x', taking the limit as
00:25:46.030 --> 00:25:47.650
'n' goes to infinity.
00:25:47.650 --> 00:25:50.100
'c' is a constant, so I
can take it outside
00:25:50.100 --> 00:25:51.550
the integral sign.
00:25:51.550 --> 00:25:54.280
Since 'c' is a constant and
it's outside the integral
00:25:54.280 --> 00:25:56.410
sign, let's look at
what 'delta x' is.
00:25:56.410 --> 00:25:59.800
'Delta x' is 'b - a' divided
by 'n', same as we were
00:25:59.800 --> 00:26:01.390
talking about earlier
in the lecture.
00:26:01.390 --> 00:26:03.070
I have 'n' of these pieces.
00:26:03.070 --> 00:26:06.370
The 'n' in the denominator
cancels the 'n' in the
00:26:06.370 --> 00:26:07.930
numerator when I add these up.
00:26:07.930 --> 00:26:10.900
And notice that this particular
sum here, no matter
00:26:10.900 --> 00:26:13.570
what 'n' is, is just 'b - a'.
00:26:13.570 --> 00:26:15.770
In other words, in the case
that 'alpha sub k' is a
00:26:15.770 --> 00:26:20.810
constant, notice that this limit
is 'c' times 'b - a'.
00:26:20.810 --> 00:26:22.370
'c' is not 0.
00:26:22.370 --> 00:26:23.570
'b' is not equal to 'a'.
00:26:23.570 --> 00:26:25.090
We have an interval here.
00:26:25.090 --> 00:26:26.820
Therefore, this will not be 0.
00:26:26.820 --> 00:26:31.510
And notice that if this is
not 0, these two things
00:26:31.510 --> 00:26:33.210
here are not equal.
00:26:33.210 --> 00:26:37.150
And by the way, the aside that
I would like to make here is
00:26:37.150 --> 00:26:43.710
that even though this error is
not negligible, notice the
00:26:43.710 --> 00:26:48.280
fact that if 'alpha sub k' is
a constant that as 'delta x'
00:26:48.280 --> 00:26:52.270
goes to 0, this whole
term will go to 0.
00:26:52.270 --> 00:26:54.460
But it doesn't go to
0 fast enough.
00:26:54.460 --> 00:26:57.760
In other words, eventually,
we're taking this sum as 'n'
00:26:57.760 --> 00:26:58.990
goes to infinity.
00:26:58.990 --> 00:27:00.460
And here's a case where, what?
00:27:00.460 --> 00:27:04.000
The pieces went to 0, but not
fast enough to become
00:27:04.000 --> 00:27:05.800
negligible.
00:27:05.800 --> 00:27:07.990
Well, let me give you something
in contrast to this.
00:27:07.990 --> 00:27:11.840
Situation number two is suppose
instead 'alpha k' is a
00:27:11.840 --> 00:27:14.120
constant times 'delta x'.
00:27:14.120 --> 00:27:16.730
'B' times 'delta x', where
'B' is a constant.
00:27:16.730 --> 00:27:20.160
In that case, notice that
summation 'k' goes from 1 to
00:27:20.160 --> 00:27:24.200
'n', 'alpha k' times 'delta x'
is just summation 'k' goes
00:27:24.200 --> 00:27:27.840
from 1 to 'n', 'B' times
''delta x' squared'.
00:27:27.840 --> 00:27:30.040
Now keep in mind again that
'delta x' is still
00:27:30.040 --> 00:27:31.560
'b - a' over 'n'.
00:27:31.560 --> 00:27:33.300
So ''delta x' squared',
of course, is 'b
00:27:33.300 --> 00:27:35.260
- a' over 'n squared'.
00:27:35.260 --> 00:27:39.070
Notice that what's inside the
summation sign here does not
00:27:39.070 --> 00:27:40.280
depend on 'k'.
00:27:40.280 --> 00:27:41.450
It's a constant.
00:27:41.450 --> 00:27:44.340
I can take it outside
the summation sign.
00:27:44.340 --> 00:27:46.580
How many terms of this
size do I have?
00:27:46.580 --> 00:27:50.360
Well, 'k' goes from 1 to 'n', so
I have 'n' of those pieces.
00:27:50.360 --> 00:27:53.350
Therefore, this sum
is given by this.
00:27:53.350 --> 00:27:55.020
This is an 'n squared' term.
00:27:55.020 --> 00:27:57.930
One of the 'n's in the
denominator cancels with my
00:27:57.930 --> 00:27:59.070
'n' in the numerator.
00:27:59.070 --> 00:28:02.550
And in this particular case, I
find that the sum, as 'k' goes
00:28:02.550 --> 00:28:06.800
from 1 to 'n', 'alpha sub k'
times 'delta x', is 'B', which
00:28:06.800 --> 00:28:10.480
is a constant, times ''b - a'
squared', which is also a
00:28:10.480 --> 00:28:13.160
constant, divided by 'n'.
00:28:13.160 --> 00:28:17.270
Now look, if I now allow 'n'
to go to infinity, my
00:28:17.270 --> 00:28:19.570
numerator is a constant.
00:28:19.570 --> 00:28:21.340
My denominator is 'n'.
00:28:21.340 --> 00:28:23.840
As 'n' goes to infinity,
my denominator
00:28:23.840 --> 00:28:25.190
increases without bound.
00:28:25.190 --> 00:28:27.070
My numerator remains constant.
00:28:27.070 --> 00:28:29.650
So the limit is 0.
00:28:29.650 --> 00:28:33.210
In other words, in the case
where 'alpha sub k' is a
00:28:33.210 --> 00:28:38.030
constant times 'delta x', this
limit is 0, the error is
00:28:38.030 --> 00:28:42.070
squeezed out, and, in this
particular case, 'w' is given
00:28:42.070 --> 00:28:46.450
by the integral from 'a' to 'b',
''g of x' dx' exactly in
00:28:46.450 --> 00:28:48.140
this particular situation.
00:28:48.140 --> 00:28:51.300
Well, the question is how many
situations shall we go through
00:28:51.300 --> 00:28:52.710
before we generalize.
00:28:52.710 --> 00:28:55.030
And the answer is since this
lecture is already becoming
00:28:55.030 --> 00:28:59.270
quite long, let's generalize now
without any more details.
00:28:59.270 --> 00:29:01.190
And the generalization
is this.
00:29:01.190 --> 00:29:05.720
In general, if you break down
'w' into increments, which
00:29:05.720 --> 00:29:09.030
we'll call 'delta 'w sub
k'', and 'delta 'w sub
00:29:09.030 --> 00:29:11.360
k'' is equal to--
00:29:11.360 --> 00:29:12.890
well, I've made a little
slip here.
00:29:12.890 --> 00:29:13.850
That should be a 'g' in here.
00:29:13.850 --> 00:29:15.480
I'm using 'g's rather
than 'f's.
00:29:15.480 --> 00:29:19.950
If 'delta 'w sub k'' is 'g of
'c sub k'' times 'delta x'
00:29:19.950 --> 00:29:24.100
plus the correction factor
'alpha k' times 'delta x',
00:29:24.100 --> 00:29:27.810
and, for each 'k', the limit
of 'alpha k' as 'delta x'
00:29:27.810 --> 00:29:29.510
approaches 0 is 0.
00:29:29.510 --> 00:29:32.800
In other words, what we're
saying is that 'alpha sub k'
00:29:32.800 --> 00:29:36.970
times 'delta x' must be a higher
order infinitesimal.
00:29:36.970 --> 00:29:40.700
If this is a higher order
infinitesimal, if 'alpha k'
00:29:40.700 --> 00:29:44.510
goes to 0 as 'delta x' goes
to 0, that says, what?
00:29:44.510 --> 00:29:48.660
That 'alpha k' times 'delta x'
is going to 0 much faster than
00:29:48.660 --> 00:29:50.100
'delta x' itself.
00:29:50.100 --> 00:29:53.570
So you compare this with our
discussion on infinitesimals
00:29:53.570 --> 00:29:55.140
earlier in our course.
00:29:55.140 --> 00:29:57.790
I think that was in block two,
but that's irrelevant here.
00:29:57.790 --> 00:30:01.370
But all I'm saying is if that
is the case, in that
00:30:01.370 --> 00:30:05.560
particular case, the limit, that
integral is exactly what
00:30:05.560 --> 00:30:06.470
we're looking for.
00:30:06.470 --> 00:30:08.350
The error has been
squeezed out.
00:30:08.350 --> 00:30:12.570
In other words, now, in
conclusion, what we must do in
00:30:12.570 --> 00:30:15.690
our present problem to answer
question number three,
00:30:15.690 --> 00:30:21.900
remember, we have approximated
delta wk by this intricate
00:30:21.900 --> 00:30:25.790
little thing, '1 + ''f prime
'c sub k' squared'
00:30:25.790 --> 00:30:26.920
times 'delta x'.
00:30:26.920 --> 00:30:29.360
In other words, in our
particular illustration in
00:30:29.360 --> 00:30:33.230
this lecture, the role of 'g'
is played by the square root
00:30:33.230 --> 00:30:35.260
of '1 + 'f prime squared''.
00:30:35.260 --> 00:30:39.750
What we must show is that this
difference is a higher order
00:30:39.750 --> 00:30:41.140
differential.
00:30:41.140 --> 00:30:45.070
And this really requires much
more advanced work than we
00:30:45.070 --> 00:30:46.430
really want to go into.
00:30:46.430 --> 00:30:49.810
The only trouble is, as a
student, I always used to be
00:30:49.810 --> 00:30:55.410
upset when the instructor said,
the proof is beyond our
00:30:55.410 --> 00:30:56.570
ability or knowledge.
00:30:56.570 --> 00:30:59.590
Whenever he used to say, the
proof is beyond our knowledge
00:30:59.590 --> 00:31:02.100
at this stage of the game, I
used to say to myself, ah, he
00:31:02.100 --> 00:31:03.190
doesn't know how to prove it.
00:31:03.190 --> 00:31:05.400
I think there's something
upsetting about this.
00:31:05.400 --> 00:31:08.250
So what I'm going to try to do
for a finale here is to at
00:31:08.250 --> 00:31:11.780
least give you a plausibility
argument that we really do
00:31:11.780 --> 00:31:15.170
squeeze the error out in
our approximation of
00:31:15.170 --> 00:31:16.410
'delta w' in this case.
00:31:16.410 --> 00:31:19.190
In other words, let me draw this
little diagram to bring
00:31:19.190 --> 00:31:21.560
in the infinitesimal
idea here.
00:31:21.560 --> 00:31:23.230
Here's my 'delta w'.
00:31:23.230 --> 00:31:24.890
Here's my 'delta s'.
00:31:24.890 --> 00:31:28.320
And what I'm doing now is I am
going to take the tangent line
00:31:28.320 --> 00:31:33.400
to the curve at 'A', use that
rather than 'delta x'.
00:31:33.400 --> 00:31:35.450
In other words, what I'm going
to say is we're going to
00:31:35.450 --> 00:31:38.630
assume that our curve doesn't
have infinite oscillations.
00:31:38.630 --> 00:31:42.200
So I can assume the special
case of a monotonically
00:31:42.200 --> 00:31:46.350
increasing function, use the
intuitive approach that in
00:31:46.350 --> 00:31:51.450
this diagram, 'delta w' is
caught between 'delta s' and
00:31:51.450 --> 00:31:57.470
'AB' plus 'BC', observing that
'BC' is just what's called
00:31:57.470 --> 00:32:02.220
'delta y' minus 'delta y-tan'.
00:32:02.220 --> 00:32:06.110
And that, by the Pythagorean
theorem, 'AB' is the square
00:32:06.110 --> 00:32:09.650
root of ''delta x' squared'
plus ''delta 'y sub tan''
00:32:09.650 --> 00:32:12.850
squared', which, of course, can
be written this particular
00:32:12.850 --> 00:32:16.230
way, namely, notice that the
slope here is the slope of
00:32:16.230 --> 00:32:21.560
this curve when 'x' is equal
to 'x sub 'k - 1''.
00:32:21.560 --> 00:32:23.900
And again, this is written out,
so I think you can fill
00:32:23.900 --> 00:32:27.490
in the details as part of your
review of the lecture and your
00:32:27.490 --> 00:32:28.420
homework assignment.
00:32:28.420 --> 00:32:31.220
All I want to do here is
present a plausibility
00:32:31.220 --> 00:32:36.170
argument using 'AB', 'AC', and
'delta s' as they occur in
00:32:36.170 --> 00:32:37.280
this diagram.
00:32:37.280 --> 00:32:40.370
All we're saying is, look, if
we're willing to make the
00:32:40.370 --> 00:32:45.000
assumption that this curve has
the right shape, 'delta w' is
00:32:45.000 --> 00:32:50.650
squeezed between 'delta
s' and 'AB' plus 'BC'.
00:32:50.650 --> 00:32:54.140
As we showed on our little inset
here, 'AB' is the square
00:32:54.140 --> 00:32:58.520
root of '1 + ''f prime'
evaluated 'x sub 'k - 1''
00:32:58.520 --> 00:33:00.600
squared' times 'delta x'.
00:33:00.600 --> 00:33:02.520
What is 'BC'?
00:33:02.520 --> 00:33:06.350
Remember, 'BC' was 'delta
y' minus 'delta y-tan' .
00:33:06.350 --> 00:33:09.410
That's just your epsilon
'delta x' of your
00:33:09.410 --> 00:33:13.510
infinitesimal idea, where the
limit of epsilon as 'delta x'
00:33:13.510 --> 00:33:15.760
approaches 0 is 0.
00:33:15.760 --> 00:33:17.860
In fact, let me just come over
here and make sure we write
00:33:17.860 --> 00:33:19.080
that part again.
00:33:19.080 --> 00:33:24.130
Remember what we saw was that
'delta y-tan' was 'dy/dx'
00:33:24.130 --> 00:33:30.320
evaluated at the point in
question plus what?
00:33:30.320 --> 00:33:33.940
An error term which was called
epsilon 'delta x', where
00:33:33.940 --> 00:33:37.160
epsilon went to 0 as 'delta
x' went to 0.
00:33:37.160 --> 00:33:39.520
And that's all I'm
saying over here.
00:33:39.520 --> 00:33:44.420
In other words, where is delta
s squeezed between right now?
00:33:44.420 --> 00:33:47.840
Well, let me put it this way,
delta s itself, by definition,
00:33:47.840 --> 00:33:49.470
is the square root of
''delta x' squared'
00:33:49.470 --> 00:33:51.030
plus ''delta y' squared'.
00:33:51.030 --> 00:33:53.010
That we saw was this.
00:33:53.010 --> 00:33:55.770
That was our beginning
definition in fact.
00:33:55.770 --> 00:33:59.600
Now if you look at our diagram
once more, notice that since
00:33:59.600 --> 00:34:04.980
our curve is always holding
water and rising, that the
00:34:04.980 --> 00:34:08.760
slope of the line 'delta
s' is greater than the
00:34:08.760 --> 00:34:11.420
slope of the line 'AB'.
00:34:11.420 --> 00:34:13.969
Putting all of this
together, we now
00:34:13.969 --> 00:34:16.050
have 'delta w' squeezed.
00:34:16.050 --> 00:34:18.440
And it was not at all trivial
in putting the
00:34:18.440 --> 00:34:20.370
squeeze on 'delta w'.
00:34:20.370 --> 00:34:23.650
There was no self-evident way
of saying just because one
00:34:23.650 --> 00:34:26.520
region was contained in another,
it must have a
00:34:26.520 --> 00:34:28.159
smaller arc length.
00:34:28.159 --> 00:34:31.730
We really had to be ingenious
in how we put the squeeze in
00:34:31.730 --> 00:34:33.110
to catch this thing.
00:34:33.110 --> 00:34:36.230
But in the long run, what we
now have shown is what?
00:34:36.230 --> 00:34:40.300
That 'delta w' is
equal to this.
00:34:40.300 --> 00:34:43.730
With an error of no greater
than epsilon 'delta x'.
00:34:43.730 --> 00:34:46.290
In other words, the exact
delta w is what?
00:34:46.290 --> 00:34:50.710
It's the square root of ''1
+ 'f prime 'x sub 'k - 1''
00:34:50.710 --> 00:34:55.429
squared' 'delta x' plus 'alpha
delta x', where alpha can be
00:34:55.429 --> 00:34:56.489
no bigger than epsilon.
00:34:56.489 --> 00:34:58.450
In other words, this is the
maximum error that we have
00:34:58.450 --> 00:35:00.050
here because it's caught
between this.
00:35:00.050 --> 00:35:06.010
Well, look, as 'delta x'
approaches 0, so does epsilon.
00:35:06.010 --> 00:35:09.570
And since alpha is no bigger
than epsilon, it must be that
00:35:09.570 --> 00:35:13.520
as 'delta x' approaches 0,
so does alpha approach 0.
00:35:13.520 --> 00:35:17.290
In other words, if we now write
'delta w' in this form,
00:35:17.290 --> 00:35:20.520
observe that, in line with what
we're saying, this is a
00:35:20.520 --> 00:35:22.800
higher order infinitesimal.
00:35:22.800 --> 00:35:27.040
And as a result, the intuitive
approach can be used as the
00:35:27.040 --> 00:35:28.710
correct answer.
00:35:28.710 --> 00:35:32.770
The idea is we could have said
earlier, look, why don't we
00:35:32.770 --> 00:35:36.390
approximate the arc length by
the straight line segment that
00:35:36.390 --> 00:35:38.490
joins the two endpoints
of the arc.
00:35:38.490 --> 00:35:40.370
And the answer is
you can do that.
00:35:40.370 --> 00:35:44.310
But you are really on shaky
grounds if you say it's
00:35:44.310 --> 00:35:46.200
self-evident that
all the error is
00:35:46.200 --> 00:35:48.020
squeezed out in the limit.
00:35:48.020 --> 00:35:49.890
This is a very, very
touchy thing.
00:35:49.890 --> 00:35:54.280
In other words, in the same way
that 0/0 is a very, very
00:35:54.280 --> 00:35:57.020
sensitive thing in the study
of differential calculus,
00:35:57.020 --> 00:35:59.350
infinity times 0 is equally as
00:35:59.350 --> 00:36:02.110
sensitive in integral calculus.
00:36:02.110 --> 00:36:04.780
The whole upshot of today's
lecture, however, is now that
00:36:04.780 --> 00:36:09.060
we've gone through this whole,
hard approach, it turns out
00:36:09.060 --> 00:36:12.250
that we can justify our
intuitive approach of
00:36:12.250 --> 00:36:15.700
approximating the arc length
by straight line segments.
00:36:15.700 --> 00:36:18.820
At any rate, this concludes
our lesson for today.
00:36:18.820 --> 00:36:20.620
And until next time, good-bye.
00:36:23.680 --> 00:36:26.880
Funding for the publication of
this video was provided by the
00:36:26.880 --> 00:36:30.930
Gabriella and Paul Rosenbaum
Foundation.
00:36:30.930 --> 00:36:35.110
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00:36:35.110 --> 00:36:39.310
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