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PROFESSOR: Hi.
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Having already studied area
and volume and its
12
00:00:36,550 --> 00:00:39,950
relationship to calculus, today,
we turn our attention
13
00:00:39,950 --> 00:00:41,330
to the study of length.
14
00:00:41,330 --> 00:00:44,860
And this may seem a bit strange
because, intuitively,
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00:00:44,860 --> 00:00:48,170
I think it's fair to assume that
you would imagine that
16
00:00:48,170 --> 00:00:51,170
length would be simpler than
area which, in turn, would be
17
00:00:51,170 --> 00:00:54,050
simpler than volume and, hence,
that perhaps we should
18
00:00:54,050 --> 00:00:56,210
have started with length
in the first place.
19
00:00:56,210 --> 00:00:59,060
The interesting thing is in
terms of our structure, which
20
00:00:59,060 --> 00:01:01,920
we so far have called
two-dimensional area,
21
00:01:01,920 --> 00:01:05,099
three-dimensional area, and
which, today, we shall call
22
00:01:05,099 --> 00:01:09,220
one-dimensional area, a rather
peculiar thing that causes a
23
00:01:09,220 --> 00:01:12,470
great deal of difficulty,
intellectually speaking,
24
00:01:12,470 --> 00:01:16,850
occurs in the study of arc
length that does not occur in
25
00:01:16,850 --> 00:01:19,990
either the study of
area or volume.
26
00:01:19,990 --> 00:01:22,760
And I think that we'll start
our investigation today
27
00:01:22,760 --> 00:01:25,040
leading up to what this
really means.
28
00:01:25,040 --> 00:01:29,190
So, as I say, I call today's
lesson 'One-dimensional Area',
29
00:01:29,190 --> 00:01:30,400
which is arc length.
30
00:01:30,400 --> 00:01:34,660
And let's show that there is a
parallel, at least in part,
31
00:01:34,660 --> 00:01:37,780
between the structure of arc
length and the structure of
32
00:01:37,780 --> 00:01:39,140
area and volume.
33
00:01:39,140 --> 00:01:43,240
You may recall that for area,
our initial axiom was that the
34
00:01:43,240 --> 00:01:45,990
building block of area
was a rectangle.
35
00:01:45,990 --> 00:01:49,490
And for volumes, the building
block we saw was a cylinder.
36
00:01:49,490 --> 00:01:50,950
For arc length--
37
00:01:50,950 --> 00:01:52,560
I think it's fairly obvious
to guess what
38
00:01:52,560 --> 00:01:53,500
we're going to say--
39
00:01:53,500 --> 00:01:56,650
the basic building block is
a straight line segment.
40
00:01:56,650 --> 00:01:59,740
And so without further ado, that
becomes our first rule,
41
00:01:59,740 --> 00:02:02,230
our first axiom, axiom
number one.
42
00:02:02,230 --> 00:02:05,870
We assume that we can measure
the length of any straight
43
00:02:05,870 --> 00:02:07,440
line segment.
44
00:02:07,440 --> 00:02:09,050
That's our building block.
45
00:02:09,050 --> 00:02:12,560
The second axiom that we assume
is that the length of
46
00:02:12,560 --> 00:02:15,140
the whole equals the sum of
the lengths of the parts.
47
00:02:15,140 --> 00:02:18,290
In other words, if an arc is
broken down into constituent
48
00:02:18,290 --> 00:02:22,080
bases, the total arc length is
equal to the sum of the arc
49
00:02:22,080 --> 00:02:23,890
lengths of the constituent
parts.
50
00:02:23,890 --> 00:02:26,670
And at this stage, we can
say, so far, so good.
51
00:02:26,670 --> 00:02:28,500
This still looks like
it's going to be the
52
00:02:28,500 --> 00:02:30,390
same as area or volume.
53
00:02:30,390 --> 00:02:34,400
But now remember what one of the
axioms for both area and
54
00:02:34,400 --> 00:02:36,080
volume were, namely, what?
55
00:02:36,080 --> 00:02:40,120
That if region 'R' was contained
in region 'S', the
56
00:02:40,120 --> 00:02:45,530
area or the volume of 'R' was
no greater than that of the
57
00:02:45,530 --> 00:02:47,410
area or volume of 'S'.
58
00:02:47,410 --> 00:02:50,120
However, for arc length,
this is not true.
59
00:02:50,120 --> 00:02:51,330
It need not be true.
60
00:02:51,330 --> 00:02:52,410
I shouldn't say it's not true.
61
00:02:52,410 --> 00:02:56,580
It need not be true that if
region 'R' is contained in 'S'
62
00:02:56,580 --> 00:03:00,680
that the perimeter of region 'R'
is less than or equal to
63
00:03:00,680 --> 00:03:01,880
the perimeter of 'S'.
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00:03:01,880 --> 00:03:05,850
In fact, this little diagram
that I've drawn over here, I
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00:03:05,850 --> 00:03:08,680
hope will show you what
I'm driving at.
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00:03:08,680 --> 00:03:11,290
Notice that it's rather clear
that the region 'R' here,
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00:03:11,290 --> 00:03:15,080
which is shaded, is contained
inside the region 'S', which
68
00:03:15,080 --> 00:03:16,370
is my rectangle.
69
00:03:16,370 --> 00:03:18,930
And yet, if you look at the
perimeter here, all these
70
00:03:18,930 --> 00:03:22,620
finger-shaped things in here, I
think it's easy to see that
71
00:03:22,620 --> 00:03:27,600
the perimeter of 'R' exceeds
the perimeter of 'S'.
72
00:03:27,600 --> 00:03:30,510
And if it's not that easy to
see, heck, just make a few
73
00:03:30,510 --> 00:03:34,180
more loops inside here and
keep wiggling this thing
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00:03:34,180 --> 00:03:37,370
around until you're convinced
that you have created this
75
00:03:37,370 --> 00:03:38,450
particular situation.
76
00:03:38,450 --> 00:03:42,790
All I want you to see here is
that it's plausible to you
77
00:03:42,790 --> 00:03:48,120
that we cannot talk about
lengths by squeezing them, as
78
00:03:48,120 --> 00:03:51,620
we did areas and volumes,
between regions that we
79
00:03:51,620 --> 00:03:54,800
already knew contained the
given region and were
80
00:03:54,800 --> 00:03:56,250
contained in the given region.
81
00:03:56,250 --> 00:03:58,910
Now let me just pause here for
one moment to make sure that
82
00:03:58,910 --> 00:04:00,140
we keep one thing straight.
83
00:04:00,140 --> 00:04:04,080
We're talking now about an
analytical approach to length.
84
00:04:04,080 --> 00:04:06,390
In other words, an approach that
will allow us to bring to
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00:04:06,390 --> 00:04:10,980
bear all of the power of
calculus to the study.
86
00:04:10,980 --> 00:04:14,380
I don't want you to forget for
a moment that intuitively, we
87
00:04:14,380 --> 00:04:17,130
certainly do know what arc
length is, just as we
88
00:04:17,130 --> 00:04:21,100
intuitively had a feeling for
what area and volume were.
89
00:04:21,100 --> 00:04:23,780
Just to freshen our memories
on this, remember the
90
00:04:23,780 --> 00:04:24,960
intuitive approach.
91
00:04:24,960 --> 00:04:30,970
That if you have an arc from 'A'
to 'B', the typical way of
92
00:04:30,970 --> 00:04:33,910
measuring the arc length is to
take, for example, a piece of
93
00:04:33,910 --> 00:04:37,480
string, lay it off along the
curve from 'A' to 'B'.
94
00:04:37,480 --> 00:04:40,430
After you've done this,
pick the string up.
95
00:04:40,430 --> 00:04:43,610
And then straighten the string
out, whatever that means, and
96
00:04:43,610 --> 00:04:45,420
measure its length
with a ruler.
97
00:04:45,420 --> 00:04:48,020
And we won't worry about how
you know whether you're
98
00:04:48,020 --> 00:04:50,490
stretching the string too
taut or what have you.
99
00:04:50,490 --> 00:04:52,890
We'll leave out these
philosophic questions.
100
00:04:52,890 --> 00:04:56,500
All we'll say is we would like
a more objective method that
101
00:04:56,500 --> 00:04:59,250
will allow us to use
mathematical analysis.
102
00:04:59,250 --> 00:05:02,420
And so what we're going to try
to do next is to find an
103
00:05:02,420 --> 00:05:07,190
analytic way that will allow us
to use calculus, but at the
104
00:05:07,190 --> 00:05:11,150
same time will give us a
definition which agrees with
105
00:05:11,150 --> 00:05:12,360
our intuition.
106
00:05:12,360 --> 00:05:14,930
And the first question is
how shall we begin.
107
00:05:14,930 --> 00:05:18,890
And as so often is the case in
mathematics, we begin our new
108
00:05:18,890 --> 00:05:23,470
quest by going back to an
old way that worked
109
00:05:23,470 --> 00:05:24,750
for a previous case.
110
00:05:24,750 --> 00:05:27,640
And hopefully, we'll find a
way of extending the old
111
00:05:27,640 --> 00:05:29,640
situation to cover the new.
112
00:05:29,640 --> 00:05:33,280
Now what does this mean in
this particular instance?
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00:05:33,280 --> 00:05:35,540
Well, let me just
call it this.
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00:05:35,540 --> 00:05:38,110
I'll call it analytical
approach, trial number one.
115
00:05:38,110 --> 00:05:42,320
What I'm going to do is try to
imitate exactly what we did in
116
00:05:42,320 --> 00:05:43,620
the area case.
117
00:05:43,620 --> 00:05:47,290
For example, if I take the
region 'R', which I'll draw
118
00:05:47,290 --> 00:05:51,160
this way here, if this is the
region 'R', namely bounded
119
00:05:51,160 --> 00:05:55,430
above by the curve 'y' equals 'f
of x', below by the x-axis,
120
00:05:55,430 --> 00:05:58,000
on the left, by the line 'x'
equals 'a', and on the right,
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00:05:58,000 --> 00:06:01,010
by the line 'x' equals 'b', how
did we find the area of
122
00:06:01,010 --> 00:06:01,860
the region 'R'?
123
00:06:01,860 --> 00:06:06,020
Well, what we did is we
inscribed and we circumscribed
124
00:06:06,020 --> 00:06:07,150
rectangles.
125
00:06:07,150 --> 00:06:10,132
And we took the limit of the
circumscribed rectangles, et
126
00:06:10,132 --> 00:06:13,270
cetera, and put the squeeze on
as 'n' went to infinity.
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00:06:13,270 --> 00:06:16,990
Now the idea is we might get the
idea that maybe we should
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00:06:16,990 --> 00:06:18,470
do the same thing
for arc length.
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00:06:18,470 --> 00:06:21,770
In other words, let me call one
of these little pieces of
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00:06:21,770 --> 00:06:23,540
arc length 'delta w'.
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00:06:23,540 --> 00:06:24,990
In other words, I'm just
isolating part
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00:06:24,990 --> 00:06:26,370
of the diagram here.
133
00:06:26,370 --> 00:06:27,640
Here's 'delta w'.
134
00:06:27,640 --> 00:06:28,770
Here's 'delta x'.
135
00:06:28,770 --> 00:06:30,420
Here's 'delta y'.
136
00:06:30,420 --> 00:06:34,810
The idea is in the same way that
I approximated a piece of
137
00:06:34,810 --> 00:06:37,830
area by an inscribed and a
circumscribed rectangle, why
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00:06:37,830 --> 00:06:41,660
can't I say something like,
well, let me let 'delta w' be
139
00:06:41,660 --> 00:06:44,410
approximately equal
to 'delta x'?
140
00:06:44,410 --> 00:06:46,820
And just to make sure that our
memories are refreshed over
141
00:06:46,820 --> 00:06:51,950
here, notice that 'delta x' is
just the length of each piece
142
00:06:51,950 --> 00:06:55,860
if the segment from 'a' to 'b',
namely of length 'b - a',
143
00:06:55,860 --> 00:06:58,670
is divided into 'n'
equal parts.
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00:06:58,670 --> 00:07:01,710
See, the idea is why can't we
mimic the same approach.
145
00:07:01,710 --> 00:07:05,140
And let me point out what is
so crucial here in terms of
146
00:07:05,140 --> 00:07:08,310
what I mentioned above, namely,
notice that the reason
147
00:07:08,310 --> 00:07:11,710
that we can say that the area of
the region 'R' is just the
148
00:07:11,710 --> 00:07:14,920
limit of 'U sub n' as 'n'
approaches infinity, where 'U
149
00:07:14,920 --> 00:07:18,000
sub n' is the area of the
circumscribed rectangles.
150
00:07:18,000 --> 00:07:21,950
The only reason we can say that
is because we squeezed 'A
151
00:07:21,950 --> 00:07:27,540
sub r' between 'L sub n', the
inscribed rectangles, and 'U
152
00:07:27,540 --> 00:07:29,990
sub n', the circumscribed
rectangles.
153
00:07:29,990 --> 00:07:32,310
And the limits of these
lower bounds and
154
00:07:32,310 --> 00:07:34,030
upper bounds were equal.
155
00:07:34,030 --> 00:07:36,830
'A sub 'r was squeezed
between these two.
156
00:07:36,830 --> 00:07:39,170
Hence, it had to equal
the common limit.
157
00:07:39,170 --> 00:07:41,370
That was the structure
that we used.
158
00:07:41,370 --> 00:07:44,910
On the other hand, we can't use
that when we're dealing
159
00:07:44,910 --> 00:07:46,060
with arc length.
160
00:07:46,060 --> 00:07:48,450
And I'll mention that in
a few moments again.
161
00:07:48,450 --> 00:07:50,860
But let me just point out what
I'm driving at this way.
162
00:07:50,860 --> 00:07:53,380
Suppose we mimic this
as we did before.
163
00:07:53,380 --> 00:07:56,620
And we say, OK, let the element
of arc length, 'delta
164
00:07:56,620 --> 00:07:59,720
w', be approximately
equal to 'delta x'.
165
00:07:59,720 --> 00:08:02,410
And now what I will do
is define script
166
00:08:02,410 --> 00:08:04,500
'L' from 'a' to 'b'.
167
00:08:04,500 --> 00:08:06,570
I don't want to call it arc
length because it may not be.
168
00:08:06,570 --> 00:08:10,090
But as a first approximation,
let me define this symbol to
169
00:08:10,090 --> 00:08:14,640
be the limit of the sum of all
these 'delta x's when we
170
00:08:14,640 --> 00:08:18,270
divide this region into 'n'
parts as 'n' goes to infinity.
171
00:08:18,270 --> 00:08:21,060
Now look, I have the right to
make up this particular
172
00:08:21,060 --> 00:08:22,540
definition.
173
00:08:22,540 --> 00:08:25,080
Now if I compute this
limit, what happens?
174
00:08:25,080 --> 00:08:28,280
Recall that we mentioned
that 'delta x' was
175
00:08:28,280 --> 00:08:30,060
'b - a' over 'n'.
176
00:08:30,060 --> 00:08:33,620
Consequently, if I have 'n' of
these pieces, the total sum
177
00:08:33,620 --> 00:08:34,490
would be what?
178
00:08:34,490 --> 00:08:37,380
'n' times b' - a' over 'n'.
179
00:08:37,380 --> 00:08:40,995
And 'n' times 'b - a' over
'n' is just 'b - a'.
180
00:08:40,995 --> 00:08:45,170
In other words, script 'L' from
'a' to 'b' is defined.
181
00:08:45,170 --> 00:08:47,880
And it's 'b - a', not 'w'.
182
00:08:47,880 --> 00:08:49,550
In other words, coming
back to our
183
00:08:49,550 --> 00:08:51,460
diagram, notice what happened.
184
00:08:51,460 --> 00:08:54,370
What we wanted was a recipe
that would give
185
00:08:54,370 --> 00:08:56,160
us this length here.
186
00:08:56,160 --> 00:09:00,600
What we found was a recipe that
gave us the length from
187
00:09:00,600 --> 00:09:01,730
'a' to 'b'.
188
00:09:01,730 --> 00:09:05,610
Now intuitively, we know that
the length from 'a' to 'b' is
189
00:09:05,610 --> 00:09:07,620
not the arc length that
we're looking for.
190
00:09:07,620 --> 00:09:10,520
In other words, what we defined
to be script 'L'
191
00:09:10,520 --> 00:09:14,290
existed as a limit, but it gave
us an answer which did
192
00:09:14,290 --> 00:09:16,530
not coincide with
our intuition.
193
00:09:16,530 --> 00:09:19,480
Since we intuitively know what
the right answer is, we must
194
00:09:19,480 --> 00:09:22,250
discard this approach in the
sense that it doesn't give us
195
00:09:22,250 --> 00:09:24,330
an answer that we have
any faith in.
196
00:09:24,330 --> 00:09:27,550
And by the way, notice where we
went wrong over here if you
197
00:09:27,550 --> 00:09:29,200
want to look at it from
that point of view.
198
00:09:29,200 --> 00:09:31,850
Notice that when we approximated
'delta w' by
199
00:09:31,850 --> 00:09:36,450
'delta x', it's clear from this
diagram that 'delta x'
200
00:09:36,450 --> 00:09:38,370
was certainly less
than 'delta w'.
201
00:09:38,370 --> 00:09:41,570
But notice that we didn't have
an upper bound here.
202
00:09:41,570 --> 00:09:45,270
Or we can make speculations
like, maybe 'delta x + delta
203
00:09:45,270 --> 00:09:48,320
y' would be more than 'delta
w', and things like this.
204
00:09:48,320 --> 00:09:50,310
We'll talk about that
more later.
205
00:09:50,310 --> 00:09:53,720
But for now, all I want us
to see is the degree of
206
00:09:53,720 --> 00:09:57,420
sophistication that enters into
the arc length problem
207
00:09:57,420 --> 00:10:00,160
that didn't bother us in either
the area or the volume
208
00:10:00,160 --> 00:10:04,670
problems, namely, we are missing
now the all-important
209
00:10:04,670 --> 00:10:06,430
squeeze element.
210
00:10:06,430 --> 00:10:09,050
Well, no sense crying
over spilt milk.
211
00:10:09,050 --> 00:10:12,560
We go on, and we try the
next type of approach.
212
00:10:12,560 --> 00:10:16,510
In other words, what we sense
now is why don't we do this.
213
00:10:16,510 --> 00:10:20,730
Instead of approximating 'delta
w' by 'delta x', why
214
00:10:20,730 --> 00:10:24,730
don't we approximate 'delta w'
by the cord that joins the two
215
00:10:24,730 --> 00:10:27,070
endpoints of the arc.
216
00:10:27,070 --> 00:10:29,070
In other words, I think that
we began to suspect
217
00:10:29,070 --> 00:10:32,530
intuitively that, somehow or
other, for a small change in
218
00:10:32,530 --> 00:10:36,200
'delta x', 'delta s' should be
a better approximation to
219
00:10:36,200 --> 00:10:38,660
'delta w' than 'delta x' was.
220
00:10:38,660 --> 00:10:41,720
Of course, the wide open
question is granted that it's
221
00:10:41,720 --> 00:10:43,710
better, is it good enough.
222
00:10:43,710 --> 00:10:46,360
Well, we'll worry about that in
a little more detail later.
223
00:10:46,360 --> 00:10:49,670
All we're saying is let 'delta
w' be approximately
224
00:10:49,670 --> 00:10:50,620
equal to 'delta s'.
225
00:10:50,620 --> 00:10:52,330
In other words, we'll
approximate 'delta
226
00:10:52,330 --> 00:10:54,100
w' by 'delta s'.
227
00:10:54,100 --> 00:10:58,230
And we'll now define 'L' from
'a' to 'b', 'L' from 'a' to
228
00:10:58,230 --> 00:11:02,160
'b' to be the limit not of the
sum of 'delta x's now, but the
229
00:11:02,160 --> 00:11:06,500
sum of the 'delta s's, as 'k'
goes from 1 to 'n', taken in
230
00:11:06,500 --> 00:11:08,790
the limit as 'n' goes
to infinity.
231
00:11:08,790 --> 00:11:11,480
And for those of us who are more
familiar with 'delta x's
232
00:11:11,480 --> 00:11:15,060
and 'delta y's, and the symbol
delta s bothers us, simply
233
00:11:15,060 --> 00:11:18,750
observe that by the Pythagorean
theorem, 'delta s'
234
00:11:18,750 --> 00:11:22,000
is related to 'delta x' and
'delta y' by ''delta s'
235
00:11:22,000 --> 00:11:23,860
squared' equals ''delta
x' squared'
236
00:11:23,860 --> 00:11:25,290
plus ''delta y' squared'.
237
00:11:25,290 --> 00:11:28,490
So we can rewrite this in
this particular form.
238
00:11:28,490 --> 00:11:32,570
In other words, I will define
capital 'L' hopefully to stand
239
00:11:32,570 --> 00:11:33,590
for length later on.
240
00:11:33,590 --> 00:11:35,490
But we'll worry about
that later too.
241
00:11:35,490 --> 00:11:39,370
But 'L' from 'a' to 'b' to
be this particular limit.
242
00:11:39,370 --> 00:11:43,200
And now I claim that there are
three natural questions with
243
00:11:43,200 --> 00:11:45,370
which we must come to grips.
244
00:11:45,370 --> 00:11:48,830
The first question is does
this limit even exist.
245
00:11:48,830 --> 00:11:50,500
Does this limit exist?
246
00:11:50,500 --> 00:11:51,920
And the answer is
that, except for
247
00:11:51,920 --> 00:11:54,200
far-fetched curves, it does.
248
00:11:54,200 --> 00:12:00,040
You really have to get a curve
that wiggles uncontrollably to
249
00:12:00,040 --> 00:12:02,930
break the possibility of
this limit existing.
250
00:12:02,930 --> 00:12:06,520
Unfortunately, there are
pathological cases, one of
251
00:12:06,520 --> 00:12:10,120
which is described in the text
assignment for this lesson, of
252
00:12:10,120 --> 00:12:13,110
a curve that doesn't have a
finite limit when you try to
253
00:12:13,110 --> 00:12:14,730
compute the arc length
this way.
254
00:12:14,730 --> 00:12:15,920
Just a little idiosyncrasy.
255
00:12:15,920 --> 00:12:19,110
However, for any curve that
comes up in real life, that
256
00:12:19,110 --> 00:12:22,250
doesn't oscillate too violently
with infinite
257
00:12:22,250 --> 00:12:25,210
variations, et cetera, et
cetera, which we won't, again,
258
00:12:25,210 --> 00:12:28,920
talk about right now, the idea
is that this limit does exist.
259
00:12:28,920 --> 00:12:31,990
As far as this course is
concerned, we shall assume the
260
00:12:31,990 --> 00:12:33,940
answer to question one is yes.
261
00:12:33,940 --> 00:12:36,520
In fact, the way we'll do it
without being dictatorial is
262
00:12:36,520 --> 00:12:38,795
we'll say, look, if this limit
doesn't exist, we just won't
263
00:12:38,795 --> 00:12:40,120
study that curve.
264
00:12:40,120 --> 00:12:42,840
In fact, we will call
a curve rectifiable
265
00:12:42,840 --> 00:12:44,510
if this limit exists.
266
00:12:44,510 --> 00:12:47,460
And so we'll assume that we
deal only with rectifiable
267
00:12:47,460 --> 00:12:50,040
curves, in other words, that
this limit does exist.
268
00:12:50,040 --> 00:12:51,380
Question number two.
269
00:12:51,380 --> 00:12:52,970
OK, the limit exists.
270
00:12:52,970 --> 00:12:54,790
So how do we compute it?
271
00:12:54,790 --> 00:12:58,060
And that, in general, is not a
very easy thing to answer.
272
00:12:58,060 --> 00:13:00,830
What's even worse though is
that after you've answered
273
00:13:00,830 --> 00:13:03,870
this, you have to come to grips
with a question that we
274
00:13:03,870 --> 00:13:07,480
were able to dodge when we
studied both area and volume,
275
00:13:07,480 --> 00:13:11,370
namely, the question is once
this limit does exist and you
276
00:13:11,370 --> 00:13:14,750
compute it, how do you know
that it agrees with our
277
00:13:14,750 --> 00:13:16,900
intuitive definition
of arc length.
278
00:13:16,900 --> 00:13:19,230
In other words, if you recall
what we did just a few minutes
279
00:13:19,230 --> 00:13:23,330
ago, we defined script 'L'
from 'a' to 'b' to be a
280
00:13:23,330 --> 00:13:24,290
certain limit.
281
00:13:24,290 --> 00:13:26,800
We showed that that
limit existed.
282
00:13:26,800 --> 00:13:30,560
The problem was is that limit,
even though it existed, did
283
00:13:30,560 --> 00:13:34,350
not give us an answer that
agreed intuitively with what
284
00:13:34,350 --> 00:13:36,700
we believed arc length
was supposed to mean.
285
00:13:36,700 --> 00:13:38,810
In other words, you see, we've
assumed the answer to the
286
00:13:38,810 --> 00:13:40,820
first question is yes.
287
00:13:40,820 --> 00:13:42,640
Now we have two questions
to answer.
288
00:13:42,640 --> 00:13:45,290
How do you compute this limit,
which is a hard question in
289
00:13:45,290 --> 00:13:46,080
it's own right?
290
00:13:46,080 --> 00:13:49,110
Secondly, once you do compute
this limit, how do you know
291
00:13:49,110 --> 00:13:52,340
that it's going to agree with
the intuitive answer that you
292
00:13:52,340 --> 00:13:53,430
get for arc length?
293
00:13:53,430 --> 00:13:56,510
And this shall be what we have
to answer in the remainder of
294
00:13:56,510 --> 00:13:58,140
our lesson today.
295
00:13:58,140 --> 00:13:59,680
Let's take these in order.
296
00:13:59,680 --> 00:14:02,370
And let's try to answer question
number two first.
297
00:14:02,370 --> 00:14:05,470
The idea is we've defined
capital 'L' from 'a' to 'b' to
298
00:14:05,470 --> 00:14:07,770
be this particular limit, and
we'd like to know if this
299
00:14:07,770 --> 00:14:09,000
limit exists.
300
00:14:09,000 --> 00:14:12,370
Not only that, but we have a
great command of calculus at
301
00:14:12,370 --> 00:14:13,590
our disposal now.
302
00:14:13,590 --> 00:14:17,490
All of the previous lessons can
be brought to bear here to
303
00:14:17,490 --> 00:14:20,470
help us put this into the
perspective of what calculus
304
00:14:20,470 --> 00:14:21,380
is all about.
305
00:14:21,380 --> 00:14:24,890
For example, when I see an
expression like this, I like
306
00:14:24,890 --> 00:14:26,670
to think in terms
of a derivative.
307
00:14:26,670 --> 00:14:29,660
A derivative reminds me of
'delta y' divided by 'delta
308
00:14:29,660 --> 00:14:30,810
x', et cetera.
309
00:14:30,810 --> 00:14:33,910
So what I do here is I factor
out a ''delta x' squared'.
310
00:14:33,910 --> 00:14:36,750
In other words, I divide through
by ''delta x' squared'
311
00:14:36,750 --> 00:14:39,390
inside the radical sign, which
is really the same
312
00:14:39,390 --> 00:14:41,920
equivalently as dividing
by 'delta x'.
313
00:14:41,920 --> 00:14:44,230
And I multiply by 'delta
x' outside.
314
00:14:44,230 --> 00:14:46,840
In other words, factoring out
with ''delta x' squared', the
315
00:14:46,840 --> 00:14:49,100
square root of ''delta x'
squared' plus ''delta y'
316
00:14:49,100 --> 00:14:53,060
squared' can be written as the
square root of '1 + ''delta y'
317
00:14:53,060 --> 00:14:56,240
over 'delta x'' squared'
times 'delta x'.
318
00:14:56,240 --> 00:14:59,900
Now the idea is that 'delta y'
over 'delta x' is the slope of
319
00:14:59,900 --> 00:15:03,910
that cord that joins the two
endpoints of 'delta w'.
320
00:15:03,910 --> 00:15:05,490
It's not a derivative
as we know it.
321
00:15:05,490 --> 00:15:07,750
It's the slope of a straight
line cord, not
322
00:15:07,750 --> 00:15:09,220
the slope of a curve.
323
00:15:09,220 --> 00:15:10,750
Now the whole idea is this.
324
00:15:10,750 --> 00:15:14,040
We know from the mean value
theorem that if our curve is
325
00:15:14,040 --> 00:15:17,820
smooth, there is a point in
the interval at which the
326
00:15:17,820 --> 00:15:20,690
derivative at that point
is equal to the
327
00:15:20,690 --> 00:15:22,020
slope of the cord.
328
00:15:22,020 --> 00:15:25,170
In other words, if 'f' is
differentiable on [a, b], we
329
00:15:25,170 --> 00:15:27,170
may invoke the Mean
Value Theorem--
330
00:15:27,170 --> 00:15:30,280
here abbreviated as MVT, the
Mean Value Theorem--
331
00:15:30,280 --> 00:15:34,660
to conclude that there is some
point 'c sub k' in our 'delta
332
00:15:34,660 --> 00:15:38,680
x' interval for which ''delta y'
over 'delta x'' is 'f prime
333
00:15:38,680 --> 00:15:39,860
of 'c sub k''.
334
00:15:39,860 --> 00:15:42,420
And in order to help you
facilitate what we're talking
335
00:15:42,420 --> 00:15:45,005
about in your minds, look at
the following diagram.
336
00:15:45,005 --> 00:15:46,110
This is all we're saying.
337
00:15:46,110 --> 00:15:48,350
What we're saying is here's
our 'delta x',
338
00:15:48,350 --> 00:15:49,520
here's our 'delta y'.
339
00:15:49,520 --> 00:15:52,870
We'll call this point 'x
sub 'k - 1'', this
340
00:15:52,870 --> 00:15:54,240
point 'x sub k'.
341
00:15:54,240 --> 00:15:56,020
This is our k-th partition.
342
00:15:56,020 --> 00:15:59,270
'Delta y' divided by 'delta
x' is just the
343
00:15:59,270 --> 00:16:00,850
slope of this line.
344
00:16:00,850 --> 00:16:03,630
See, that's just the
slope of this line.
345
00:16:03,630 --> 00:16:05,825
And what the Mean Value Theorem
says is if this curve
346
00:16:05,825 --> 00:16:10,280
is smooth, some place on this
arc, there is a point where
347
00:16:10,280 --> 00:16:16,380
the line tangent to the curve
is parallel to this cord.
348
00:16:16,380 --> 00:16:18,980
And that's what I'm calling
the point 'c sub k'.
349
00:16:18,980 --> 00:16:23,170
'c sub k' is the point at which
the slope of the curve
350
00:16:23,170 --> 00:16:25,430
is equal to the slope
of the cord.
351
00:16:25,430 --> 00:16:29,290
In other words, if 'f' is
continuous, I can conclude
352
00:16:29,290 --> 00:16:34,880
that 'L' from 'a' to 'b' is the
limit as 'n' approaches
353
00:16:34,880 --> 00:16:39,500
infinity, summation 'k' goes
from 1 to 'n', square root of
354
00:16:39,500 --> 00:16:44,830
'1 + ''f prime 'c sub k''
squared' times 'delta x'.
355
00:16:44,830 --> 00:16:49,650
And notice that this now
starts to look like my
356
00:16:49,650 --> 00:16:52,530
definite integral according to
the definition that we were
357
00:16:52,530 --> 00:16:55,720
talking about in our earlier
lectures in this block.
358
00:16:55,720 --> 00:16:59,070
In fact, how can we invoke the
first fundamental theorem of
359
00:16:59,070 --> 00:17:00,340
integral calculus?
360
00:17:00,340 --> 00:17:04,583
Remember, if this expression
here-- it's
361
00:17:04,583 --> 00:17:05,829
not an integral yet--
362
00:17:05,829 --> 00:17:08,589
happens to be a continuous
function, then we're in pretty
363
00:17:08,589 --> 00:17:09,700
good shape.
364
00:17:09,700 --> 00:17:12,630
In other words, if I can assume
that 'f prime' is
365
00:17:12,630 --> 00:17:13,240
continuous--
366
00:17:13,240 --> 00:17:15,470
let's go over here and
continue on here.
367
00:17:15,470 --> 00:17:20,660
See, what I'm saying is if I can
assume that 'f prime' is
368
00:17:20,660 --> 00:17:24,200
continuous, well, look, the
square of a continuous
369
00:17:24,200 --> 00:17:25,980
function is continuous.
370
00:17:25,980 --> 00:17:29,280
The sum of two continuous
functions is continuous.
371
00:17:29,280 --> 00:17:31,900
And the square root of a
continuous function is
372
00:17:31,900 --> 00:17:32,830
continuous.
373
00:17:32,830 --> 00:17:36,150
In other words, and this is a
key point, if the derivative
374
00:17:36,150 --> 00:17:41,380
is continuous, I can conclude
that the 'L' from 'a' to 'b'
375
00:17:41,380 --> 00:17:43,650
can be replaced by the definite
integral from 'a' to
376
00:17:43,650 --> 00:17:48,690
'b' square root of '1 + ''dy/dx'
squared'' times 'dx',
377
00:17:48,690 --> 00:17:54,150
which I quickly point out
may be hard to evaluate.
378
00:17:54,150 --> 00:17:57,370
In other words, one thing I
could try to do over here is
379
00:17:57,370 --> 00:18:01,110
to find the function g whose
derivative with respect to 'x'
380
00:18:01,110 --> 00:18:04,000
is the square root of '1 +
''dy/dx' squared'' and
381
00:18:04,000 --> 00:18:05,890
evaluate that between
'a' and 'b'.
382
00:18:05,890 --> 00:18:10,850
I can put approximations on
here, whatever I want.
383
00:18:10,850 --> 00:18:12,500
In fact, let's summarize
it down here.
384
00:18:12,500 --> 00:18:16,220
If 'f' is differentiable on the
closed interval from 'a'
385
00:18:16,220 --> 00:18:20,000
to 'b' and if 'f prime'
is the derivative--
386
00:18:20,000 --> 00:18:21,320
you see, 'f prime'--
387
00:18:21,320 --> 00:18:23,750
is also continuous on the closed
interval from 'a' to
388
00:18:23,750 --> 00:18:27,250
'b', then not only does capital
'L' from 'a' to 'b'
389
00:18:27,250 --> 00:18:30,410
exist, but it's given
computationally by this
390
00:18:30,410 --> 00:18:32,150
particular integral.
391
00:18:32,150 --> 00:18:35,220
And that answers question number
two, that the limit
392
00:18:35,220 --> 00:18:38,000
exists, and this is what
it's equal to.
393
00:18:38,000 --> 00:18:40,110
The problem that we're
faced with--
394
00:18:40,110 --> 00:18:41,180
and I've written this out.
395
00:18:41,180 --> 00:18:44,010
I think it looks harder
than what it says.
396
00:18:44,010 --> 00:18:46,825
But I've taken the trouble to
write this whole thing out, so
397
00:18:46,825 --> 00:18:49,870
that if you have trouble
following what I'm saying,
398
00:18:49,870 --> 00:18:52,530
that you can see this thing
blocked out for you.
399
00:18:52,530 --> 00:18:53,590
The idea is this.
400
00:18:53,590 --> 00:18:57,060
What we have done is we have
approximated 'delta
401
00:18:57,060 --> 00:18:59,170
w' by 'delta s'.
402
00:18:59,170 --> 00:19:02,390
Then what we said is 'w'
is the sum of all
403
00:19:02,390 --> 00:19:03,900
these 'delta w's.
404
00:19:03,900 --> 00:19:08,460
And since each 'delta w' is
approximately 'delta s', then
405
00:19:08,460 --> 00:19:12,340
what we can be sure of is that
'w' is approximated by this
406
00:19:12,340 --> 00:19:13,730
sum over here.
407
00:19:13,730 --> 00:19:14,740
Now here's what we did.
408
00:19:14,740 --> 00:19:17,050
We didn't work with 'w'
at all after this.
409
00:19:17,050 --> 00:19:20,000
We turned our attention
to this.
410
00:19:20,000 --> 00:19:21,440
This is what we did
in our case here.
411
00:19:21,440 --> 00:19:24,750
And we showed that this
limit existed.
412
00:19:24,750 --> 00:19:29,810
We showed that the limit, as
'k' went from 1 to 'n' and
413
00:19:29,810 --> 00:19:33,790
then went to infinity of these
pieces here, was 'L' of 'ab'.
414
00:19:33,790 --> 00:19:34,770
And that existed.
415
00:19:34,770 --> 00:19:39,600
What we did not show is that
this limit was w itself.
416
00:19:39,600 --> 00:19:42,940
Intuitively, you might say, if
I put the squeeze on this,
417
00:19:42,940 --> 00:19:45,670
doesn't this get rid of
all the error for me?
418
00:19:45,670 --> 00:19:48,930
We haven't shown that we've
gotten rid of all the error.
419
00:19:48,930 --> 00:19:51,750
In essence, how do we know
if all the error has been
420
00:19:51,750 --> 00:19:53,000
squeezed out?
421
00:19:53,000 --> 00:19:56,210
This is precisely what question
three is all about.
422
00:19:56,210 --> 00:19:59,530
Again, going back to what we did
earlier, remember, when we
423
00:19:59,530 --> 00:20:03,330
approximated 'delta w' by 'delta
x', then we said, OK,
424
00:20:03,330 --> 00:20:06,380
add up all these 'delta x's, and
take the limit as 'n' goes
425
00:20:06,380 --> 00:20:07,190
to infinity.
426
00:20:07,190 --> 00:20:11,090
We found that that limit was
'b - a', which was not the
427
00:20:11,090 --> 00:20:12,670
length of the curve.
428
00:20:12,670 --> 00:20:14,750
In other words, somehow or
other, even though the limit
429
00:20:14,750 --> 00:20:17,760
existed, we did not squeeze
out all the error.
430
00:20:17,760 --> 00:20:21,220
And this is why the study of
arc length is so difficult.
431
00:20:21,220 --> 00:20:23,210
Because we don't have a
sandwiching effect.
432
00:20:23,210 --> 00:20:27,210
It is very difficult for us to
figure out when we've squeezed
433
00:20:27,210 --> 00:20:28,350
out all the error.
434
00:20:28,350 --> 00:20:30,290
So at any rate, let
me generalize
435
00:20:30,290 --> 00:20:32,880
question number three.
436
00:20:32,880 --> 00:20:34,120
Remember what question
number three is?
437
00:20:34,120 --> 00:20:36,510
How do we know that if
the limit exists,
438
00:20:36,510 --> 00:20:37,610
it's equal to 'w'?
439
00:20:37,610 --> 00:20:40,860
All I'm saying is don't even
worry about arc length.
440
00:20:40,860 --> 00:20:44,420
Just suppose that 'w' is any
function defined on a closed
441
00:20:44,420 --> 00:20:47,470
interval from 'a' to 'b' and
that we've approximated 'delta
442
00:20:47,470 --> 00:20:52,350
w' by something of the form 'g
of 'c sub k'' times 'delta x',
443
00:20:52,350 --> 00:20:55,700
where 'g' is what I call
some intuitive function
444
00:20:55,700 --> 00:20:57,140
defined on [a, b].
445
00:20:57,140 --> 00:21:01,160
For example, in our earlier
example, we started with
446
00:21:01,160 --> 00:21:02,820
'delta w' being arc length.
447
00:21:02,820 --> 00:21:06,460
And we approximated 'delta w' by
'delta x' in which case 'g'
448
00:21:06,460 --> 00:21:09,220
would've been the function
which is identically 1.
449
00:21:09,220 --> 00:21:12,130
In the area situation, remember
we approximated
450
00:21:12,130 --> 00:21:15,120
'delta A' by something
times 'delta x'.
451
00:21:15,120 --> 00:21:16,860
Well, what times 'delta x'?
452
00:21:16,860 --> 00:21:18,570
Well, it was the height
of a rectangle.
453
00:21:18,570 --> 00:21:21,370
In other words, we look at the
thing we're trying to find, we
454
00:21:21,370 --> 00:21:22,370
use our intuition--
455
00:21:22,370 --> 00:21:25,090
and this is difficult because
intuition varies from person
456
00:21:25,090 --> 00:21:25,890
to person--
457
00:21:25,890 --> 00:21:28,290
and we say, what would make
a good approximation here.
458
00:21:28,290 --> 00:21:30,600
What would be an
approximation?
459
00:21:30,600 --> 00:21:35,770
We say, OK, let's approximate
'delta w' by 'g of 'c sub k''
460
00:21:35,770 --> 00:21:38,420
times 'delta x', where 'c'
is some point in the
461
00:21:38,420 --> 00:21:39,570
interval, et cetera.
462
00:21:39,570 --> 00:21:42,290
Then we add up all of these
'delta w's as 'k'
463
00:21:42,290 --> 00:21:43,800
goes from 1 to 'n'.
464
00:21:43,800 --> 00:21:45,830
We say, OK, that's
approximately
465
00:21:45,830 --> 00:21:47,420
this thing over here.
466
00:21:47,420 --> 00:21:51,750
Now what we have shown is that
if 'g' is continuous on [a, b]
467
00:21:51,750 --> 00:21:57,200
then as 'n' goes to infinity,
this particular limit exists
468
00:21:57,200 --> 00:21:59,210
and is denoted by the
integral from 'a' to
469
00:21:59,210 --> 00:22:01,840
'b', ''g of x' dx'.
470
00:22:01,840 --> 00:22:03,900
This is what we've
shown so far.
471
00:22:03,900 --> 00:22:07,320
What the big question is is,
granted that this limit
472
00:22:07,320 --> 00:22:09,810
exists, does it equal 'w'?
473
00:22:09,810 --> 00:22:13,280
In other words, is 'w' equal
to the integral from 'a' to
474
00:22:13,280 --> 00:22:15,870
b', ''g of x' dx'?
475
00:22:15,870 --> 00:22:19,060
That's what the remainder of
today's lesson is about as far
476
00:22:19,060 --> 00:22:20,400
as arc length is concerned.
477
00:22:20,400 --> 00:22:23,170
And I'm going to solve this
problem in general first and
478
00:22:23,170 --> 00:22:25,420
then make some applications
about this
479
00:22:25,420 --> 00:22:27,590
to arc length itself.
480
00:22:27,590 --> 00:22:30,560
And by the way, what we're going
to see next is you may
481
00:22:30,560 --> 00:22:33,610
remember that very, very early
in our course, we came to
482
00:22:33,610 --> 00:22:36,220
grips with something called
infinitesimals.
483
00:22:36,220 --> 00:22:40,630
We came to grips with this delta
y tan infinitesimals of
484
00:22:40,630 --> 00:22:41,520
higher order.
485
00:22:41,520 --> 00:22:45,120
And now we're going to see how
just as this came up in
486
00:22:45,120 --> 00:22:49,170
differential calculus, these
same problems of approximation
487
00:22:49,170 --> 00:22:51,200
come up in integral calculus.
488
00:22:51,200 --> 00:22:54,020
The only difference, as we've
mentioned before, is instead
489
00:22:54,020 --> 00:22:56,690
of having to come to grips with
the indeterminate form
490
00:22:56,690 --> 00:22:59,890
0/0, we're going to have to
come to grips with the
491
00:22:59,890 --> 00:23:02,700
indeterminate form
infinity times 0.
492
00:23:02,700 --> 00:23:04,560
Let me show you what
I mean by that.
493
00:23:04,560 --> 00:23:05,550
The idea is this.
494
00:23:05,550 --> 00:23:08,920
Let's suppose that our case
'delta w'-- we've broken up
495
00:23:08,920 --> 00:23:10,800
'w' now into increments--
496
00:23:10,800 --> 00:23:13,960
and let's suppose that we're
approximating 'delta w', as we
497
00:23:13,960 --> 00:23:18,500
said before, by 'g of 'c sub
k'' times 'delta x'.
498
00:23:18,500 --> 00:23:20,800
Well, what do we mean by we're
approximating this?
499
00:23:20,800 --> 00:23:23,400
What we mean is there's
some error in here.
500
00:23:23,400 --> 00:23:27,110
Let's call the error 'alpha
sub k' times 'delta x'.
501
00:23:27,110 --> 00:23:29,960
In other words, this is just
a correction factor.
502
00:23:29,960 --> 00:23:32,260
This is what we have to
add on to this to make
503
00:23:32,260 --> 00:23:34,130
this equality whole.
504
00:23:34,130 --> 00:23:37,510
Once I add on the error, I'm
no longer working with an
505
00:23:37,510 --> 00:23:38,230
inequality.
506
00:23:38,230 --> 00:23:40,920
I'm working with an equality.
507
00:23:40,920 --> 00:23:43,130
And that allows me to
use some theorems.
508
00:23:43,130 --> 00:23:47,140
What I can say now is by
definition, w is the sum of
509
00:23:47,140 --> 00:23:48,720
all these 'delta w's.
510
00:23:48,720 --> 00:23:52,120
But 'delta w' being a sum, we
can use theorems about the
511
00:23:52,120 --> 00:23:53,390
sigma notation.
512
00:23:53,390 --> 00:23:56,080
In other words, what is the sum
of all these 'delta w's?
513
00:23:56,080 --> 00:23:59,200
It's the sum of all of these
pieces plus the sum of all of
514
00:23:59,200 --> 00:24:01,700
these pieces, which I've
written over here.
515
00:24:01,700 --> 00:24:05,130
And now you see, if I transpose,
I get that 'w'
516
00:24:05,130 --> 00:24:09,330
minus this sum is equal to the
'sum k' goes from 1 to 'n',
517
00:24:09,330 --> 00:24:12,260
'alpha k' times 'delta x'.
518
00:24:12,260 --> 00:24:14,670
Now the next thing I do
is take the limit
519
00:24:14,670 --> 00:24:16,600
as 'n' goes to infinity.
520
00:24:16,600 --> 00:24:20,810
By definition, since 'g' is a
continuous function, this
521
00:24:20,810 --> 00:24:23,350
limit here is just the definite
integral from 'a' to
522
00:24:23,350 --> 00:24:25,530
'b', ''g of x' dx'.
523
00:24:25,530 --> 00:24:28,490
On the other hand, this limit
here is what we have to
524
00:24:28,490 --> 00:24:29,560
investigate.
525
00:24:29,560 --> 00:24:32,500
In other words, we would like
to know whether 'w' is equal
526
00:24:32,500 --> 00:24:34,010
to the definite integral
or not.
527
00:24:34,010 --> 00:24:36,910
If we look at this particular
equation, what we have now
528
00:24:36,910 --> 00:24:40,350
shown is whatever the
relationship is between these
529
00:24:40,350 --> 00:24:44,130
two terms, it's typified by the
fact that this difference
530
00:24:44,130 --> 00:24:45,810
is this particular limit.
531
00:24:45,810 --> 00:24:48,840
In other words, if this limit
happens to be 0, then the
532
00:24:48,840 --> 00:24:52,330
integral will equal what we're
setting out to show it's equal
533
00:24:52,330 --> 00:24:53,850
to, namely, this function
itself.
534
00:24:53,850 --> 00:24:57,450
On the other hand, what we're
saying is we do not know that
535
00:24:57,450 --> 00:24:58,770
this limit is 0.
536
00:24:58,770 --> 00:25:00,470
By the way, notice what's
happening over here.
537
00:25:00,470 --> 00:25:04,020
As 'n' goes to infinity, 'delta
x' is going to 0.
538
00:25:04,020 --> 00:25:07,260
In other words, each individual
term in the sum is
539
00:25:07,260 --> 00:25:10,540
going to 0, but the number of
pieces is becoming infinite.
540
00:25:10,540 --> 00:25:13,340
There's your infinity
times 0 form here.
541
00:25:13,340 --> 00:25:16,420
And let me show you a case where
the pieces are growing
542
00:25:16,420 --> 00:25:20,170
too fast in number to be offset
by the fact that their
543
00:25:20,170 --> 00:25:21,730
size is going to 0.
544
00:25:21,730 --> 00:25:25,110
For the sake of argument, let me
suppose that 'alpha sub k'
545
00:25:25,110 --> 00:25:29,100
happens to be some non-0
constant for all 'k'.
546
00:25:29,100 --> 00:25:33,110
If I come back to this
expression here, if 'alpha sub
547
00:25:33,110 --> 00:25:36,310
k' is equal to a constant, I'll
replace 'alpha sub k' by
548
00:25:36,310 --> 00:25:38,300
that constant, which is 'c'.
549
00:25:38,300 --> 00:25:39,450
I now have what?
550
00:25:39,450 --> 00:25:42,340
That the limit that I'm looking
for is the 'sum k'
551
00:25:42,340 --> 00:25:46,030
goes from 1 to 'n', 'c' times
'delta x', taking the limit as
552
00:25:46,030 --> 00:25:47,650
'n' goes to infinity.
553
00:25:47,650 --> 00:25:50,100
'c' is a constant, so I
can take it outside
554
00:25:50,100 --> 00:25:51,550
the integral sign.
555
00:25:51,550 --> 00:25:54,280
Since 'c' is a constant and
it's outside the integral
556
00:25:54,280 --> 00:25:56,410
sign, let's look at
what 'delta x' is.
557
00:25:56,410 --> 00:25:59,800
'Delta x' is 'b - a' divided
by 'n', same as we were
558
00:25:59,800 --> 00:26:01,390
talking about earlier
in the lecture.
559
00:26:01,390 --> 00:26:03,070
I have 'n' of these pieces.
560
00:26:03,070 --> 00:26:06,370
The 'n' in the denominator
cancels the 'n' in the
561
00:26:06,370 --> 00:26:07,930
numerator when I add these up.
562
00:26:07,930 --> 00:26:10,900
And notice that this particular
sum here, no matter
563
00:26:10,900 --> 00:26:13,570
what 'n' is, is just 'b - a'.
564
00:26:13,570 --> 00:26:15,770
In other words, in the case
that 'alpha sub k' is a
565
00:26:15,770 --> 00:26:20,810
constant, notice that this limit
is 'c' times 'b - a'.
566
00:26:20,810 --> 00:26:22,370
'c' is not 0.
567
00:26:22,370 --> 00:26:23,570
'b' is not equal to 'a'.
568
00:26:23,570 --> 00:26:25,090
We have an interval here.
569
00:26:25,090 --> 00:26:26,820
Therefore, this will not be 0.
570
00:26:26,820 --> 00:26:31,510
And notice that if this is
not 0, these two things
571
00:26:31,510 --> 00:26:33,210
here are not equal.
572
00:26:33,210 --> 00:26:37,150
And by the way, the aside that
I would like to make here is
573
00:26:37,150 --> 00:26:43,710
that even though this error is
not negligible, notice the
574
00:26:43,710 --> 00:26:48,280
fact that if 'alpha sub k' is
a constant that as 'delta x'
575
00:26:48,280 --> 00:26:52,270
goes to 0, this whole
term will go to 0.
576
00:26:52,270 --> 00:26:54,460
But it doesn't go to
0 fast enough.
577
00:26:54,460 --> 00:26:57,760
In other words, eventually,
we're taking this sum as 'n'
578
00:26:57,760 --> 00:26:58,990
goes to infinity.
579
00:26:58,990 --> 00:27:00,460
And here's a case where, what?
580
00:27:00,460 --> 00:27:04,000
The pieces went to 0, but not
fast enough to become
581
00:27:04,000 --> 00:27:05,800
negligible.
582
00:27:05,800 --> 00:27:07,990
Well, let me give you something
in contrast to this.
583
00:27:07,990 --> 00:27:11,840
Situation number two is suppose
instead 'alpha k' is a
584
00:27:11,840 --> 00:27:14,120
constant times 'delta x'.
585
00:27:14,120 --> 00:27:16,730
'B' times 'delta x', where
'B' is a constant.
586
00:27:16,730 --> 00:27:20,160
In that case, notice that
summation 'k' goes from 1 to
587
00:27:20,160 --> 00:27:24,200
'n', 'alpha k' times 'delta x'
is just summation 'k' goes
588
00:27:24,200 --> 00:27:27,840
from 1 to 'n', 'B' times
''delta x' squared'.
589
00:27:27,840 --> 00:27:30,040
Now keep in mind again that
'delta x' is still
590
00:27:30,040 --> 00:27:31,560
'b - a' over 'n'.
591
00:27:31,560 --> 00:27:33,300
So ''delta x' squared',
of course, is 'b
592
00:27:33,300 --> 00:27:35,260
- a' over 'n squared'.
593
00:27:35,260 --> 00:27:39,070
Notice that what's inside the
summation sign here does not
594
00:27:39,070 --> 00:27:40,280
depend on 'k'.
595
00:27:40,280 --> 00:27:41,450
It's a constant.
596
00:27:41,450 --> 00:27:44,340
I can take it outside
the summation sign.
597
00:27:44,340 --> 00:27:46,580
How many terms of this
size do I have?
598
00:27:46,580 --> 00:27:50,360
Well, 'k' goes from 1 to 'n', so
I have 'n' of those pieces.
599
00:27:50,360 --> 00:27:53,350
Therefore, this sum
is given by this.
600
00:27:53,350 --> 00:27:55,020
This is an 'n squared' term.
601
00:27:55,020 --> 00:27:57,930
One of the 'n's in the
denominator cancels with my
602
00:27:57,930 --> 00:27:59,070
'n' in the numerator.
603
00:27:59,070 --> 00:28:02,550
And in this particular case, I
find that the sum, as 'k' goes
604
00:28:02,550 --> 00:28:06,800
from 1 to 'n', 'alpha sub k'
times 'delta x', is 'B', which
605
00:28:06,800 --> 00:28:10,480
is a constant, times ''b - a'
squared', which is also a
606
00:28:10,480 --> 00:28:13,160
constant, divided by 'n'.
607
00:28:13,160 --> 00:28:17,270
Now look, if I now allow 'n'
to go to infinity, my
608
00:28:17,270 --> 00:28:19,570
numerator is a constant.
609
00:28:19,570 --> 00:28:21,340
My denominator is 'n'.
610
00:28:21,340 --> 00:28:23,840
As 'n' goes to infinity,
my denominator
611
00:28:23,840 --> 00:28:25,190
increases without bound.
612
00:28:25,190 --> 00:28:27,070
My numerator remains constant.
613
00:28:27,070 --> 00:28:29,650
So the limit is 0.
614
00:28:29,650 --> 00:28:33,210
In other words, in the case
where 'alpha sub k' is a
615
00:28:33,210 --> 00:28:38,030
constant times 'delta x', this
limit is 0, the error is
616
00:28:38,030 --> 00:28:42,070
squeezed out, and, in this
particular case, 'w' is given
617
00:28:42,070 --> 00:28:46,450
by the integral from 'a' to 'b',
''g of x' dx' exactly in
618
00:28:46,450 --> 00:28:48,140
this particular situation.
619
00:28:48,140 --> 00:28:51,300
Well, the question is how many
situations shall we go through
620
00:28:51,300 --> 00:28:52,710
before we generalize.
621
00:28:52,710 --> 00:28:55,030
And the answer is since this
lecture is already becoming
622
00:28:55,030 --> 00:28:59,270
quite long, let's generalize now
without any more details.
623
00:28:59,270 --> 00:29:01,190
And the generalization
is this.
624
00:29:01,190 --> 00:29:05,720
In general, if you break down
'w' into increments, which
625
00:29:05,720 --> 00:29:09,030
we'll call 'delta 'w sub
k'', and 'delta 'w sub
626
00:29:09,030 --> 00:29:11,360
k'' is equal to--
627
00:29:11,360 --> 00:29:12,890
well, I've made a little
slip here.
628
00:29:12,890 --> 00:29:13,850
That should be a 'g' in here.
629
00:29:13,850 --> 00:29:15,480
I'm using 'g's rather
than 'f's.
630
00:29:15,480 --> 00:29:19,950
If 'delta 'w sub k'' is 'g of
'c sub k'' times 'delta x'
631
00:29:19,950 --> 00:29:24,100
plus the correction factor
'alpha k' times 'delta x',
632
00:29:24,100 --> 00:29:27,810
and, for each 'k', the limit
of 'alpha k' as 'delta x'
633
00:29:27,810 --> 00:29:29,510
approaches 0 is 0.
634
00:29:29,510 --> 00:29:32,800
In other words, what we're
saying is that 'alpha sub k'
635
00:29:32,800 --> 00:29:36,970
times 'delta x' must be a higher
order infinitesimal.
636
00:29:36,970 --> 00:29:40,700
If this is a higher order
infinitesimal, if 'alpha k'
637
00:29:40,700 --> 00:29:44,510
goes to 0 as 'delta x' goes
to 0, that says, what?
638
00:29:44,510 --> 00:29:48,660
That 'alpha k' times 'delta x'
is going to 0 much faster than
639
00:29:48,660 --> 00:29:50,100
'delta x' itself.
640
00:29:50,100 --> 00:29:53,570
So you compare this with our
discussion on infinitesimals
641
00:29:53,570 --> 00:29:55,140
earlier in our course.
642
00:29:55,140 --> 00:29:57,790
I think that was in block two,
but that's irrelevant here.
643
00:29:57,790 --> 00:30:01,370
But all I'm saying is if that
is the case, in that
644
00:30:01,370 --> 00:30:05,560
particular case, the limit, that
integral is exactly what
645
00:30:05,560 --> 00:30:06,470
we're looking for.
646
00:30:06,470 --> 00:30:08,350
The error has been
squeezed out.
647
00:30:08,350 --> 00:30:12,570
In other words, now, in
conclusion, what we must do in
648
00:30:12,570 --> 00:30:15,690
our present problem to answer
question number three,
649
00:30:15,690 --> 00:30:21,900
remember, we have approximated
delta wk by this intricate
650
00:30:21,900 --> 00:30:25,790
little thing, '1 + ''f prime
'c sub k' squared'
651
00:30:25,790 --> 00:30:26,920
times 'delta x'.
652
00:30:26,920 --> 00:30:29,360
In other words, in our
particular illustration in
653
00:30:29,360 --> 00:30:33,230
this lecture, the role of 'g'
is played by the square root
654
00:30:33,230 --> 00:30:35,260
of '1 + 'f prime squared''.
655
00:30:35,260 --> 00:30:39,750
What we must show is that this
difference is a higher order
656
00:30:39,750 --> 00:30:41,140
differential.
657
00:30:41,140 --> 00:30:45,070
And this really requires much
more advanced work than we
658
00:30:45,070 --> 00:30:46,430
really want to go into.
659
00:30:46,430 --> 00:30:49,810
The only trouble is, as a
student, I always used to be
660
00:30:49,810 --> 00:30:55,410
upset when the instructor said,
the proof is beyond our
661
00:30:55,410 --> 00:30:56,570
ability or knowledge.
662
00:30:56,570 --> 00:30:59,590
Whenever he used to say, the
proof is beyond our knowledge
663
00:30:59,590 --> 00:31:02,100
at this stage of the game, I
used to say to myself, ah, he
664
00:31:02,100 --> 00:31:03,190
doesn't know how to prove it.
665
00:31:03,190 --> 00:31:05,400
I think there's something
upsetting about this.
666
00:31:05,400 --> 00:31:08,250
So what I'm going to try to do
for a finale here is to at
667
00:31:08,250 --> 00:31:11,780
least give you a plausibility
argument that we really do
668
00:31:11,780 --> 00:31:15,170
squeeze the error out in
our approximation of
669
00:31:15,170 --> 00:31:16,410
'delta w' in this case.
670
00:31:16,410 --> 00:31:19,190
In other words, let me draw this
little diagram to bring
671
00:31:19,190 --> 00:31:21,560
in the infinitesimal
idea here.
672
00:31:21,560 --> 00:31:23,230
Here's my 'delta w'.
673
00:31:23,230 --> 00:31:24,890
Here's my 'delta s'.
674
00:31:24,890 --> 00:31:28,320
And what I'm doing now is I am
going to take the tangent line
675
00:31:28,320 --> 00:31:33,400
to the curve at 'A', use that
rather than 'delta x'.
676
00:31:33,400 --> 00:31:35,450
In other words, what I'm going
to say is we're going to
677
00:31:35,450 --> 00:31:38,630
assume that our curve doesn't
have infinite oscillations.
678
00:31:38,630 --> 00:31:42,200
So I can assume the special
case of a monotonically
679
00:31:42,200 --> 00:31:46,350
increasing function, use the
intuitive approach that in
680
00:31:46,350 --> 00:31:51,450
this diagram, 'delta w' is
caught between 'delta s' and
681
00:31:51,450 --> 00:31:57,470
'AB' plus 'BC', observing that
'BC' is just what's called
682
00:31:57,470 --> 00:32:02,220
'delta y' minus 'delta y-tan'.
683
00:32:02,220 --> 00:32:06,110
And that, by the Pythagorean
theorem, 'AB' is the square
684
00:32:06,110 --> 00:32:09,650
root of ''delta x' squared'
plus ''delta 'y sub tan''
685
00:32:09,650 --> 00:32:12,850
squared', which, of course, can
be written this particular
686
00:32:12,850 --> 00:32:16,230
way, namely, notice that the
slope here is the slope of
687
00:32:16,230 --> 00:32:21,560
this curve when 'x' is equal
to 'x sub 'k - 1''.
688
00:32:21,560 --> 00:32:23,900
And again, this is written out,
so I think you can fill
689
00:32:23,900 --> 00:32:27,490
in the details as part of your
review of the lecture and your
690
00:32:27,490 --> 00:32:28,420
homework assignment.
691
00:32:28,420 --> 00:32:31,220
All I want to do here is
present a plausibility
692
00:32:31,220 --> 00:32:36,170
argument using 'AB', 'AC', and
'delta s' as they occur in
693
00:32:36,170 --> 00:32:37,280
this diagram.
694
00:32:37,280 --> 00:32:40,370
All we're saying is, look, if
we're willing to make the
695
00:32:40,370 --> 00:32:45,000
assumption that this curve has
the right shape, 'delta w' is
696
00:32:45,000 --> 00:32:50,650
squeezed between 'delta
s' and 'AB' plus 'BC'.
697
00:32:50,650 --> 00:32:54,140
As we showed on our little inset
here, 'AB' is the square
698
00:32:54,140 --> 00:32:58,520
root of '1 + ''f prime'
evaluated 'x sub 'k - 1''
699
00:32:58,520 --> 00:33:00,600
squared' times 'delta x'.
700
00:33:00,600 --> 00:33:02,520
What is 'BC'?
701
00:33:02,520 --> 00:33:06,350
Remember, 'BC' was 'delta
y' minus 'delta y-tan' .
702
00:33:06,350 --> 00:33:09,410
That's just your epsilon
'delta x' of your
703
00:33:09,410 --> 00:33:13,510
infinitesimal idea, where the
limit of epsilon as 'delta x'
704
00:33:13,510 --> 00:33:15,760
approaches 0 is 0.
705
00:33:15,760 --> 00:33:17,860
In fact, let me just come over
here and make sure we write
706
00:33:17,860 --> 00:33:19,080
that part again.
707
00:33:19,080 --> 00:33:24,130
Remember what we saw was that
'delta y-tan' was 'dy/dx'
708
00:33:24,130 --> 00:33:30,320
evaluated at the point in
question plus what?
709
00:33:30,320 --> 00:33:33,940
An error term which was called
epsilon 'delta x', where
710
00:33:33,940 --> 00:33:37,160
epsilon went to 0 as 'delta
x' went to 0.
711
00:33:37,160 --> 00:33:39,520
And that's all I'm
saying over here.
712
00:33:39,520 --> 00:33:44,420
In other words, where is delta
s squeezed between right now?
713
00:33:44,420 --> 00:33:47,840
Well, let me put it this way,
delta s itself, by definition,
714
00:33:47,840 --> 00:33:49,470
is the square root of
''delta x' squared'
715
00:33:49,470 --> 00:33:51,030
plus ''delta y' squared'.
716
00:33:51,030 --> 00:33:53,010
That we saw was this.
717
00:33:53,010 --> 00:33:55,770
That was our beginning
definition in fact.
718
00:33:55,770 --> 00:33:59,600
Now if you look at our diagram
once more, notice that since
719
00:33:59,600 --> 00:34:04,980
our curve is always holding
water and rising, that the
720
00:34:04,980 --> 00:34:08,760
slope of the line 'delta
s' is greater than the
721
00:34:08,760 --> 00:34:11,420
slope of the line 'AB'.
722
00:34:11,420 --> 00:34:13,969
Putting all of this
together, we now
723
00:34:13,969 --> 00:34:16,050
have 'delta w' squeezed.
724
00:34:16,050 --> 00:34:18,440
And it was not at all trivial
in putting the
725
00:34:18,440 --> 00:34:20,370
squeeze on 'delta w'.
726
00:34:20,370 --> 00:34:23,650
There was no self-evident way
of saying just because one
727
00:34:23,650 --> 00:34:26,520
region was contained in another,
it must have a
728
00:34:26,520 --> 00:34:28,159
smaller arc length.
729
00:34:28,159 --> 00:34:31,730
We really had to be ingenious
in how we put the squeeze in
730
00:34:31,730 --> 00:34:33,110
to catch this thing.
731
00:34:33,110 --> 00:34:36,230
But in the long run, what we
now have shown is what?
732
00:34:36,230 --> 00:34:40,300
That 'delta w' is
equal to this.
733
00:34:40,300 --> 00:34:43,730
With an error of no greater
than epsilon 'delta x'.
734
00:34:43,730 --> 00:34:46,290
In other words, the exact
delta w is what?
735
00:34:46,290 --> 00:34:50,710
It's the square root of ''1
+ 'f prime 'x sub 'k - 1''
736
00:34:50,710 --> 00:34:55,429
squared' 'delta x' plus 'alpha
delta x', where alpha can be
737
00:34:55,429 --> 00:34:56,489
no bigger than epsilon.
738
00:34:56,489 --> 00:34:58,450
In other words, this is the
maximum error that we have
739
00:34:58,450 --> 00:35:00,050
here because it's caught
between this.
740
00:35:00,050 --> 00:35:06,010
Well, look, as 'delta x'
approaches 0, so does epsilon.
741
00:35:06,010 --> 00:35:09,570
And since alpha is no bigger
than epsilon, it must be that
742
00:35:09,570 --> 00:35:13,520
as 'delta x' approaches 0,
so does alpha approach 0.
743
00:35:13,520 --> 00:35:17,290
In other words, if we now write
'delta w' in this form,
744
00:35:17,290 --> 00:35:20,520
observe that, in line with what
we're saying, this is a
745
00:35:20,520 --> 00:35:22,800
higher order infinitesimal.
746
00:35:22,800 --> 00:35:27,040
And as a result, the intuitive
approach can be used as the
747
00:35:27,040 --> 00:35:28,710
correct answer.
748
00:35:28,710 --> 00:35:32,770
The idea is we could have said
earlier, look, why don't we
749
00:35:32,770 --> 00:35:36,390
approximate the arc length by
the straight line segment that
750
00:35:36,390 --> 00:35:38,490
joins the two endpoints
of the arc.
751
00:35:38,490 --> 00:35:40,370
And the answer is
you can do that.
752
00:35:40,370 --> 00:35:44,310
But you are really on shaky
grounds if you say it's
753
00:35:44,310 --> 00:35:46,200
self-evident that
all the error is
754
00:35:46,200 --> 00:35:48,020
squeezed out in the limit.
755
00:35:48,020 --> 00:35:49,890
This is a very, very
touchy thing.
756
00:35:49,890 --> 00:35:54,280
In other words, in the same way
that 0/0 is a very, very
757
00:35:54,280 --> 00:35:57,020
sensitive thing in the study
of differential calculus,
758
00:35:57,020 --> 00:35:59,350
infinity times 0 is equally as
759
00:35:59,350 --> 00:36:02,110
sensitive in integral calculus.
760
00:36:02,110 --> 00:36:04,780
The whole upshot of today's
lecture, however, is now that
761
00:36:04,780 --> 00:36:09,060
we've gone through this whole,
hard approach, it turns out
762
00:36:09,060 --> 00:36:12,250
that we can justify our
intuitive approach of
763
00:36:12,250 --> 00:36:15,700
approximating the arc length
by straight line segments.
764
00:36:15,700 --> 00:36:18,820
At any rate, this concludes
our lesson for today.
765
00:36:18,820 --> 00:36:20,620
And until next time, good-bye.
766
00:36:20,620 --> 00:36:23,680
767
00:36:23,680 --> 00:36:26,880
Funding for the publication of
this video was provided by the
768
00:36:26,880 --> 00:36:30,930
Gabriella and Paul Rosenbaum
Foundation.
769
00:36:30,930 --> 00:36:35,110
Help OCW continue to provide
free and open access to MIT
770
00:36:35,110 --> 00:36:39,310
courses by making a donation
at ocw.mit.edu/donate.
771
00:36:39,310 --> 00:36:44,052