1
00:00:00,000 --> 00:00:00,510


2
00:00:00,510 --> 00:00:03,759
If we're just dealing with two dimensions, and we want to find

3
00:00:03,759 --> 00:00:07,459
the area under a curve, we have good tools in our toolkit

4
00:00:07,459 --> 00:00:11,050
already to do it, and I'll just remind us of our tools.

5
00:00:11,050 --> 00:00:14,429
so let's say, that's the x-axis, that's the y-axis, let

6
00:00:14,429 --> 00:00:18,449
me draw some arbitrary function right here, and that's

7
00:00:18,449 --> 00:00:20,129
my function f of x.

8
00:00:20,129 --> 00:00:25,210
And let's say we want to find the area between x is equal to

9
00:00:25,210 --> 00:00:29,810
a, so that's x equal to a, and x is equal to b.

10
00:00:29,809 --> 00:00:32,140
We saw this many, many, many videos ago.

11
00:00:32,140 --> 00:00:35,009
The way you can think about it, is you take super small widths

12
00:00:35,009 --> 00:00:37,609
of x, or super small changes in x.

13
00:00:37,609 --> 00:00:40,350
We could call them delta x's, but because they're so small,

14
00:00:40,350 --> 00:00:43,649
we're going to call them a dx.

15
00:00:43,649 --> 00:00:46,299
Super, infinitesimally small changes in x.

16
00:00:46,299 --> 00:00:48,779
And then you multiply them times the value of f

17
00:00:48,780 --> 00:00:50,300
of x at that point.

18
00:00:50,299 --> 00:00:52,709
So you multiply it times the height at that point, which

19
00:00:52,710 --> 00:00:54,730
is the value of f of x.

20
00:00:54,729 --> 00:01:01,209
So you get f of x times each of these infinitesimally small

21
00:01:01,210 --> 00:01:04,930
bases, that'll give you the area of this infinitesimally

22
00:01:04,930 --> 00:01:07,500
narrow rectangle right there.

23
00:01:07,500 --> 00:01:10,219
And since each of these guys are infinitely small, you're

24
00:01:10,219 --> 00:01:12,370
going to have an infinite number of these rectangles

25
00:01:12,370 --> 00:01:14,050
in order to fill the space.

26
00:01:14,049 --> 00:01:15,480
You're going to have an infinite number

27
00:01:15,480 --> 00:01:16,880
of these, right?

28
00:01:16,879 --> 00:01:20,250
And so the tool we use was the definite integral.

29
00:01:20,250 --> 00:01:23,969
The definite integral is a sum, is an infinite sum of these

30
00:01:23,969 --> 00:01:27,000
infinitely small areas, or these infinitely

31
00:01:27,000 --> 00:01:27,909
small rectangles.

32
00:01:27,909 --> 00:01:31,789
And the notations that we use, they would go from a b.

33
00:01:31,790 --> 00:01:34,690
And we've done many videos on how do you evaluate

34
00:01:34,689 --> 00:01:35,230
these things.

35
00:01:35,230 --> 00:01:37,010
I just want to remind you, conceptually,

36
00:01:37,010 --> 00:01:38,210
what this is saying.

37
00:01:38,209 --> 00:01:42,000
This is conceptually saying, let's take a small change in x,

38
00:01:42,000 --> 00:01:47,620
multiply it times the height at that point, and you're going to

39
00:01:47,620 --> 00:01:49,740
have an infinite number of these, because these x's are

40
00:01:49,739 --> 00:01:52,109
super small, they're infinitely small, so you're going to have

41
00:01:52,109 --> 00:01:53,329
an infinite number of those.

42
00:01:53,329 --> 00:01:55,879
So take an infinite sum of all of those, from x is equal

43
00:01:55,879 --> 00:01:57,629
to a to x is equal to b.

44
00:01:57,629 --> 00:02:00,149
And that's just our standard definite integral.

45
00:02:00,150 --> 00:02:03,230
Now what I want to do in this video is extend this, broaden

46
00:02:03,230 --> 00:02:06,320
this a little bit, to solve, I guess it maybe could say a

47
00:02:06,319 --> 00:02:09,239
harder or a broader class of problems.

48
00:02:09,240 --> 00:02:13,379
Let's say that we are, let's go to three dimensions now.

49
00:02:13,379 --> 00:02:14,969
And I'll just draw the x-y plane first.

50
00:02:14,969 --> 00:02:19,819
Maybe I'll keep this, just to kind of make the analogy clear.

51
00:02:19,819 --> 00:02:21,449
I'm going to kind of flatten this, so we

52
00:02:21,449 --> 00:02:22,459
have some perspective.

53
00:02:22,460 --> 00:02:25,490
So let's say that this right here is the y-axis, kind of

54
00:02:25,490 --> 00:02:27,750
going behind the screen.

55
00:02:27,750 --> 00:02:29,409
You can imagine if I just pushed on this

56
00:02:29,409 --> 00:02:30,599
and knocked it down.

57
00:02:30,599 --> 00:02:37,479
So that's the y-axis, and that is my x-axis right there.

58
00:02:37,479 --> 00:02:41,629
And let's say I some path in the x-y plane.

59
00:02:41,629 --> 00:02:44,460
And in order to really define a path in the x-y plane, I'll

60
00:02:44,460 --> 00:02:50,560
have to parameterize both the x and y variables.

61
00:02:50,560 --> 00:02:55,120
So let's say that x is equal to, let me switch colors.

62
00:02:55,120 --> 00:02:57,349
I'm using that orange too much.

63
00:02:57,349 --> 00:03:03,049
Let's say that x is is equal to some function of some parameter

64
00:03:03,050 --> 00:03:08,450
t, and let's say y is equal to some other function of that

65
00:03:08,449 --> 00:03:11,319
same parameter t, and let's say we're going to start, we're

66
00:03:11,319 --> 00:03:15,090
going to have t go from, t is going to be greater than or

67
00:03:15,090 --> 00:03:18,840
equal to a, and then less than or equal to b.

68
00:03:18,840 --> 00:03:22,879
Now this will define a path in the x-y plane, and if this

69
00:03:22,879 --> 00:03:25,889
seems confusing, you might want to review the videos on

70
00:03:25,889 --> 00:03:27,429
parametric equations.

71
00:03:27,430 --> 00:03:32,060
But essentially, when t is equal to a, you're going to

72
00:03:32,060 --> 00:03:35,319
have x is equal to, so t is equal to a, you're going to

73
00:03:35,319 --> 00:03:39,629
have x is equal to g of a, and you're going to have

74
00:03:39,629 --> 00:03:43,165
y is equal to h of a.

75
00:03:43,165 --> 00:03:45,579
So you're going to have this point right here, so maybe it

76
00:03:45,580 --> 00:03:49,760
might be, I don't know, I'll just draw a random point here.

77
00:03:49,759 --> 00:03:52,969
When t is equal to a, you're going to plot the

78
00:03:52,969 --> 00:03:54,590
coordinate point g of a.

79
00:03:54,590 --> 00:03:56,270
That's going to be our x-coordinate.

80
00:03:56,270 --> 00:03:58,030
This is g of a, right here.

81
00:03:58,030 --> 00:04:01,319
And then our y-coordinate is going to be h of a.

82
00:04:01,319 --> 00:04:01,519
Right?

83
00:04:01,520 --> 00:04:03,930
You just put t is equal to a in each of these equations,

84
00:04:03,930 --> 00:04:05,430
and then you get a value for x and y.

85
00:04:05,430 --> 00:04:08,450
So this coordinate right here would be h of a.

86
00:04:08,449 --> 00:04:10,939
And then, you would keep incrementing t larger and

87
00:04:10,939 --> 00:04:13,189
larger, until you get to b, but you're going to get a series of

88
00:04:13,189 --> 00:04:17,189
points that are going to look something like that.

89
00:04:17,189 --> 00:04:22,410
That right there is a curve, or it's a path, in the x-y plane.

90
00:04:22,410 --> 00:04:24,430
And you know, you're saying, how does that relate

91
00:04:24,430 --> 00:04:25,430
to that right now?

92
00:04:25,430 --> 00:04:26,250
What are we doing?

93
00:04:26,250 --> 00:04:28,000
Well, let me just write a c here, for saying, that's our

94
00:04:28,000 --> 00:04:29,750
curve, our that's our path.

95
00:04:29,750 --> 00:04:34,089
Now, let's say I have another function that associates every

96
00:04:34,089 --> 00:04:37,009
point in the x-y plane with some value.

97
00:04:37,009 --> 00:04:42,480
So let's say I have some function, f of x y.

98
00:04:42,480 --> 00:04:44,800
What it does is associate every point on the x-y

99
00:04:44,800 --> 00:04:46,170
plane with some value.

100
00:04:46,170 --> 00:04:47,890
So let me plot f of x y.

101
00:04:47,889 --> 00:04:50,360
Let me make a vertical axis here.

102
00:04:50,360 --> 00:04:53,460
We could do a different color.

103
00:04:53,459 --> 00:04:56,089
Call it the f of x y axis, maybe we could even call it

104
00:04:56,089 --> 00:04:58,009
the z-axis, if you want to.

105
00:04:58,009 --> 00:05:00,079
But some vertical axis right there.

106
00:05:00,079 --> 00:05:04,050
And for every point, so if you give me an x and a y, and put

107
00:05:04,050 --> 00:05:07,860
into my f of x y function, it's going to give you some point.

108
00:05:07,860 --> 00:05:11,160
So I can just draw some type of a surface that

109
00:05:11,160 --> 00:05:12,375
f of x y represents.

110
00:05:12,375 --> 00:05:14,920
And this'll all become a lot more concrete when I do

111
00:05:14,920 --> 00:05:16,439
some concrete examples.

112
00:05:16,439 --> 00:05:18,949
So let's say that f of x y looks something like this.

113
00:05:18,949 --> 00:05:20,889
I'm going to try my best to draw it.

114
00:05:20,889 --> 00:05:23,439
I'll do a different color.

115
00:05:23,439 --> 00:05:25,089
Let's say f of x y.

116
00:05:25,089 --> 00:05:25,729
Some surface.

117
00:05:25,730 --> 00:05:26,795
I'll draw part of it.

118
00:05:26,795 --> 00:05:29,230
It's some surface that looks, let's say it looks

119
00:05:29,230 --> 00:05:31,220
something like that.

120
00:05:31,220 --> 00:05:34,680
That is f of x y.

121
00:05:34,680 --> 00:05:38,310
And remember, all this is, is you give me an x, you give me a

122
00:05:38,310 --> 00:05:41,800
y, you pop it into f of x y, it's going to give me some

123
00:05:41,800 --> 00:05:44,060
third value that we're going to plot in this vertical

124
00:05:44,060 --> 00:05:44,780
axis right here.

125
00:05:44,779 --> 00:05:46,759
I mean, example, f of x y?

126
00:05:46,759 --> 00:05:49,110
It could be, I'm not saying this is a particular case,

127
00:05:49,110 --> 00:05:50,650
it could be x plus y.

128
00:05:50,649 --> 00:05:52,299
It could be f of x y.

129
00:05:52,300 --> 00:05:53,199
These are just examples.

130
00:05:53,199 --> 00:05:55,750
It could be x times y.

131
00:05:55,750 --> 00:05:59,350
If x is 1, y is 2, f of x y will be 1 times 2.

132
00:05:59,350 --> 00:06:03,110
But let's say when you plot, for every point on the x-y

133
00:06:03,110 --> 00:06:05,569
plane, when you plot f of x y you get this surface up

134
00:06:05,569 --> 00:06:08,599
here, and we want to do something interesting.

135
00:06:08,600 --> 00:06:12,129
We want to figure out, not the area under this curve, this

136
00:06:12,129 --> 00:06:14,230
was very simple when we did it the first time.

137
00:06:14,230 --> 00:06:19,640
I want to find the area if you imagine a curtain, or a fence,

138
00:06:19,639 --> 00:06:22,569
that goes along this curve.

139
00:06:22,569 --> 00:06:24,750
You can imagine this being a very straight linear path,

140
00:06:24,750 --> 00:06:27,639
going just along the x-axis from a to b.

141
00:06:27,639 --> 00:06:30,329
Now we have this kind of crazy, curvy path that's going

142
00:06:30,329 --> 00:06:32,069
along the x-y plane.

143
00:06:32,069 --> 00:06:35,029
And you can imagine if you drew a wall, or curtain, or a fence

144
00:06:35,029 --> 00:06:39,750
that went straight up from this to my f of x y, let me do my

145
00:06:39,750 --> 00:06:42,100
best effort to draw that.

146
00:06:42,100 --> 00:06:42,790
Let me draw it.

147
00:06:42,790 --> 00:06:45,550
So it's going to go up to there, and maybe this point

148
00:06:45,550 --> 00:06:47,139
corresponds to there.

149
00:06:47,139 --> 00:06:49,479
And when you draw that curtain up, it's going to intersect

150
00:06:49,480 --> 00:06:51,720
it something like that.

151
00:06:51,720 --> 00:06:52,940
Let's say it looks something like that.

152
00:06:52,939 --> 00:06:56,550
So this point right here corresponds to that

153
00:06:56,550 --> 00:06:58,079
point right there.

154
00:06:58,079 --> 00:07:01,629
So if you imagine, you have a curtain, f of x y is the roof,

155
00:07:01,629 --> 00:07:04,620
and this is a, what I've drawn here, this curve, this kind of

156
00:07:04,620 --> 00:07:07,600
shows you the bottom of a wall.

157
00:07:07,600 --> 00:07:09,660
This is some kind of crazy wall.

158
00:07:09,660 --> 00:07:13,250
And let me say, this point it corresponds to, well, actually,

159
00:07:13,250 --> 00:07:15,204
let me draw it little bit different.

160
00:07:15,204 --> 00:07:18,620


161
00:07:18,620 --> 00:07:22,269
This point will correspond to some point up here, so when you

162
00:07:22,269 --> 00:07:24,799
trace where it intersects, it will look something maybe

163
00:07:24,800 --> 00:07:26,520
like that, I don't know.

164
00:07:26,519 --> 00:07:27,409
Something like that.

165
00:07:27,410 --> 00:07:31,220
And I'm trying my best to help you visualize this.

166
00:07:31,220 --> 00:07:34,836
So maybe I'll shade this in to make it a little solid,

167
00:07:34,836 --> 00:07:36,830
let's say f of x y is little transparent.

168
00:07:36,829 --> 00:07:37,500
You can see.

169
00:07:37,500 --> 00:07:40,360
But you have this curvy-looking wall here.

170
00:07:40,360 --> 00:07:43,879
And the whole point of this video is, how can we figure out

171
00:07:43,879 --> 00:07:49,889
the area of this curvy-looking wall, that's essentially the

172
00:07:49,889 --> 00:07:52,474
wall or the fence that happens if you go from this curve

173
00:07:52,475 --> 00:07:56,165
and jump up, and hit the ceiling at this f of x y?

174
00:07:56,165 --> 00:07:58,769
So let's think a little bit about how we can do it.

175
00:07:58,769 --> 00:08:01,469
Well, if we just use the analogy of what we did

176
00:08:01,470 --> 00:08:02,860
previously, we could say, well look.

177
00:08:02,860 --> 00:08:06,740


178
00:08:06,740 --> 00:08:11,120
Let's make a little change in distance of our curve.

179
00:08:11,120 --> 00:08:13,889
Let's call that ds.

180
00:08:13,889 --> 00:08:15,740
That's a little change in distance of my

181
00:08:15,740 --> 00:08:17,060
curve, right there.

182
00:08:17,060 --> 00:08:20,199
And if I multiply that change in distance of the curve times

183
00:08:20,199 --> 00:08:29,269
f of x y at that point, I'm going to get the area of that

184
00:08:29,269 --> 00:08:30,859
little rectangle right there.

185
00:08:30,860 --> 00:08:31,189
Right?

186
00:08:31,189 --> 00:08:34,549
So if I take the ds, my change in my, you can imagine the arc

187
00:08:34,549 --> 00:08:38,389
length of this curve at that point, so let me write, you

188
00:08:38,389 --> 00:08:51,330
know, ds is equal to super small change in arc length of

189
00:08:51,330 --> 00:08:52,879
our path, or of our curve.

190
00:08:52,879 --> 00:08:54,200
That's our ds.

191
00:08:54,200 --> 00:08:57,340
So you can imagine, the area of that little rectangle right

192
00:08:57,340 --> 00:09:02,639
there, along my curvy wall, is going to be ds, I'll make it a

193
00:09:02,639 --> 00:09:07,000
capital S, ds times the height at that point.

194
00:09:07,000 --> 00:09:10,759
Well, that's f of x y.

195
00:09:10,759 --> 00:09:13,289
And then if I take the sum, because these are infinitely

196
00:09:13,289 --> 00:09:16,980
narrow, these ds's have infinitely small width, if I

197
00:09:16,980 --> 00:09:21,860
were take the infinite sum of all of those guys, from t is

198
00:09:21,860 --> 00:09:28,129
equal to a to t is equal to b, right, from t is equal to a, I

199
00:09:28,129 --> 00:09:30,610
keep taking the sum of those rectangles, to t is equal

200
00:09:30,610 --> 00:09:34,110
to b, right there, that will give me my area.

201
00:09:34,110 --> 00:09:36,370
I'm just using the exact same logic as I did up there.

202
00:09:36,370 --> 00:09:38,659
I'm not being very mathematically rigorous, but I

203
00:09:38,659 --> 00:09:40,689
want to give you the intuition of what we're doing.

204
00:09:40,690 --> 00:09:44,190
We're really just bending the base of this thing to get a

205
00:09:44,190 --> 00:09:47,360
curvy wall instead of a straight, direct wall

206
00:09:47,360 --> 00:09:48,269
like we had up here.

207
00:09:48,269 --> 00:09:51,329
But you're saying, Sal, this is all abstract, and how can I

208
00:09:51,330 --> 00:09:53,800
even calculate something like this, this makes no sense to

209
00:09:53,799 --> 00:09:57,219
me, I have an s here, I have an x and a y, I have a t,

210
00:09:57,220 --> 00:09:58,519
what can I do with this?

211
00:09:58,519 --> 00:10:00,850
And let's see if we can make some headway.

212
00:10:00,850 --> 00:10:03,440
And I promise you, when we do it with a tangible problem,

213
00:10:03,440 --> 00:10:05,420
the end product of this video is going to be a little

214
00:10:05,419 --> 00:10:07,409
bit hairy to look at.

215
00:10:07,409 --> 00:10:10,029
But when we do it with an actual problem, it'll actually,

216
00:10:10,029 --> 00:10:13,309
I think, be very concrete, and you'll see it's not too

217
00:10:13,309 --> 00:10:14,059
hard to deal with.

218
00:10:14,059 --> 00:10:17,789
But let's see if we can get all of this in terms of t.

219
00:10:17,789 --> 00:10:21,389
So first of all, let's focus just on this ds.

220
00:10:21,389 --> 00:10:25,069
So let me re-pick up the x-y axis.

221
00:10:25,070 --> 00:10:27,910
So if I were to reflip the x-y, let me switch colors,

222
00:10:27,909 --> 00:10:30,319
this is just getting a little monotonous.

223
00:10:30,320 --> 00:10:33,805
So if I were to reflip the x-y axis like that, actually, let

224
00:10:33,804 --> 00:10:35,009
me do that with that same green, so you know we're

225
00:10:35,009 --> 00:10:38,000
dealing with the same x-y axis.

226
00:10:38,000 --> 00:10:42,340
So that's my y-axis, that is my x-axis.

227
00:10:42,340 --> 00:10:45,550
And so this path right here, if I were to just draw it straight

228
00:10:45,549 --> 00:10:48,044
up like this, it would look something like this.

229
00:10:48,044 --> 00:10:51,059


230
00:10:51,059 --> 00:10:51,319
Right?

231
00:10:51,320 --> 00:10:54,040
That's my path, my arc.

232
00:10:54,039 --> 00:10:56,929
You know, this is when t is equal to a, so this is t is

233
00:10:56,929 --> 00:11:00,089
equal to a, this is t is equal to b.

234
00:11:00,090 --> 00:11:01,905
Same thing, I just kind of picked it back up so

235
00:11:01,904 --> 00:11:03,110
you can visualize it.

236
00:11:03,110 --> 00:11:07,039
And we say that we have some change in arc length, let's

237
00:11:07,039 --> 00:11:08,879
say, let me switch colors.

238
00:11:08,879 --> 00:11:11,210
Let's say that this one right here.

239
00:11:11,210 --> 00:11:13,470
Let's say that's some small change in arc length, and

240
00:11:13,470 --> 00:11:15,639
we're calling that ds.

241
00:11:15,639 --> 00:11:19,199
Now, is there some way to relate ds to infinitely

242
00:11:19,200 --> 00:11:21,710
small changes in x or y?

243
00:11:21,710 --> 00:11:24,730
Well, if we think about it, if we really-- and this is all a

244
00:11:24,730 --> 00:11:26,930
little bit hand-wavy, I'm not being mathematically rigorous,

245
00:11:26,929 --> 00:11:29,899
but I think it'll give you the correct intuition-- if you

246
00:11:29,899 --> 00:11:33,159
imagine this is, you can figure out the length of ds if you

247
00:11:33,159 --> 00:11:37,809
know the length of these super small changes in x and

248
00:11:37,809 --> 00:11:40,799
super small changes in y.

249
00:11:40,799 --> 00:11:43,949
So if this distance right here is ds, infinitesimally small

250
00:11:43,950 --> 00:11:46,640
change in x, this distance right here is dy,

251
00:11:46,639 --> 00:11:50,090
infinitesimally small change in y, right?

252
00:11:50,090 --> 00:11:53,210
Then we could figure out ds from the Pythagorean Theorem.

253
00:11:53,210 --> 00:11:57,660
You can say that ds is going to be, it's the hypotenuse of this

254
00:11:57,659 --> 00:12:04,309
triang.e It's equal to the square root of dx squared

255
00:12:04,309 --> 00:12:09,919
plus dy squared.

256
00:12:09,919 --> 00:12:13,089
So that seems to make things a little bit, you know, we can

257
00:12:13,090 --> 00:12:14,830
get rid of the ds all of a sudden.

258
00:12:14,830 --> 00:12:18,740
So let's rewrite this little expression here, using this

259
00:12:18,740 --> 00:12:21,850
sense of what ds, is really the square root of dx

260
00:12:21,850 --> 00:12:23,159
squared plus dy squared.

261
00:12:23,159 --> 00:12:25,059
And I'm not being very rigorous, and actually it's

262
00:12:25,059 --> 00:12:27,359
very hard to be rigorous with differentials, but intuitively

263
00:12:27,360 --> 00:12:29,029
I think it makes a lot of sense.

264
00:12:29,029 --> 00:12:32,199
So we can say that this integral, the area of this

265
00:12:32,200 --> 00:12:36,080
curvy curtain, is going to be the integral from t is equal to

266
00:12:36,080 --> 00:12:44,389
a to t is equal to b of f of x y, instead of writing ds, we

267
00:12:44,389 --> 00:12:50,980
can write this, times the square root of dx squared

268
00:12:50,980 --> 00:12:52,860
plus dy squared.

269
00:12:52,860 --> 00:12:55,289
Now we at least got rid of this big capital S, but we still

270
00:12:55,289 --> 00:12:58,009
haven't solved the problem of, how do you solve something, you

271
00:12:58,009 --> 00:13:00,090
know, an integral, a definite integral that looks like this?

272
00:13:00,090 --> 00:13:02,290
We have it in terms of t here, but we only have it in

273
00:13:02,289 --> 00:13:04,409
terms of x's and y's here.

274
00:13:04,409 --> 00:13:06,490
So we need to get everything in terms of t.

275
00:13:06,490 --> 00:13:09,710
Well, we know x and y are both functions of t, so we can

276
00:13:09,710 --> 00:13:11,530
actually rewrite it like this.

277
00:13:11,529 --> 00:13:18,220
We can rewrite it as from t is equal to a, to t is equal to b.

278
00:13:18,220 --> 00:13:21,800
And f of x y, we can write it, f is a function of x, which is

279
00:13:21,799 --> 00:13:26,629
a function of t, and f is also a function of y, which is

280
00:13:26,629 --> 00:13:28,509
also a function of t.

281
00:13:28,509 --> 00:13:31,759
So you give me a t, I'll be able to give you an x or y, and

282
00:13:31,759 --> 00:13:34,629
once you give me an x or y, I can figure out what f is.

283
00:13:34,629 --> 00:13:38,320
So we have that, and then we have this part right here.

284
00:13:38,320 --> 00:13:40,100
I'll do it in orange.

285
00:13:40,100 --> 00:13:45,379
Square root of dx squared plus dy squared.

286
00:13:45,379 --> 00:13:49,000
But we still don't have things in terms of t.

287
00:13:49,000 --> 00:13:51,360
We need a dt someplace here in order be able to

288
00:13:51,360 --> 00:13:52,460
evaluate this integral.

289
00:13:52,460 --> 00:13:54,019
And we'll see that in the next video, when I do

290
00:13:54,019 --> 00:13:54,889
a concrete problem.

291
00:13:54,889 --> 00:13:58,559
But I really want to give you a sense for the end product, the

292
00:13:58,559 --> 00:14:00,569
formula we're going to get at the end product of this

293
00:14:00,570 --> 00:14:02,070
video, where it comes from.

294
00:14:02,070 --> 00:14:05,710
So one thing we can do, is if we allow ourselves to

295
00:14:05,710 --> 00:14:09,180
algebraically manipulate differentials, what we can do

296
00:14:09,179 --> 00:14:13,120
is let us multiply and divide by dt.

297
00:14:13,120 --> 00:14:16,509
So one way to think about it, you could rewrite, so let me

298
00:14:16,509 --> 00:14:19,129
just do this orange part right here.

299
00:14:19,129 --> 00:14:20,360
Let's do a little side right here.

300
00:14:20,360 --> 00:14:23,860
So if you take this orange part, and write it in pink, and

301
00:14:23,860 --> 00:14:29,460
you have dx squared, and then you have plus dy squared, and

302
00:14:29,460 --> 00:14:34,040
let's say you just multiply it times dt over dt, right?

303
00:14:34,039 --> 00:14:36,879
That's a small change in t, divided by a small change in t.

304
00:14:36,879 --> 00:14:39,500
That's 1, so of course you can multiply it by that.

305
00:14:39,500 --> 00:14:43,639
If we're to bring in this part inside of the square root sign,

306
00:14:43,639 --> 00:14:45,169
right, so let me rewrite this.

307
00:14:45,169 --> 00:14:51,209
This is the same thing as 1 over dt times the square root

308
00:14:51,210 --> 00:14:57,129
of dx squared plus dy squared, and then times that dt.

309
00:14:57,129 --> 00:14:57,419
Right?

310
00:14:57,419 --> 00:14:58,669
I just wanted to write it this way to show you I'm

311
00:14:58,669 --> 00:15:00,039
just multiplying by 1.

312
00:15:00,039 --> 00:15:02,740
And here, I'm just taking this dt, writing it there, and

313
00:15:02,740 --> 00:15:04,190
leaving this over here.

314
00:15:04,190 --> 00:15:07,130
And now if I wanted to bring this into the square root sign,

315
00:15:07,129 --> 00:15:10,509
this is the same thing, this is equal to, and I'll do it very

316
00:15:10,509 --> 00:15:13,789
slowly, just to make sure, I'll allow you to believe that I'm

317
00:15:13,789 --> 00:15:15,649
not doing anything shady with the algebra.

318
00:15:15,649 --> 00:15:18,759
This is the same thing as the square root of 1 over dt

319
00:15:18,759 --> 00:15:21,519
squared, let me make the radical a little bit bigger,

320
00:15:21,519 --> 00:15:27,980
times dx squared plus dy squared, and all of

321
00:15:27,980 --> 00:15:30,409
that times dt, right?

322
00:15:30,409 --> 00:15:32,250
I didn't do anything, you could just take the square root of

323
00:15:32,250 --> 00:15:34,080
this and you'd get 1 over dt.

324
00:15:34,080 --> 00:15:38,680
And if I just distribute this, this is equal to the square

325
00:15:38,679 --> 00:15:46,769
root, and we have our dt at the end, of dx squared, or we could

326
00:15:46,769 --> 00:15:55,699
even write, dx over dt squared, plus dy over dt squared.

327
00:15:55,700 --> 00:15:58,590
Right? dx squared over dt squared is just dx over dt

328
00:15:58,590 --> 00:16:00,840
squared, same thing with the y's.

329
00:16:00,840 --> 00:16:03,100
And now all of a sudden, this starts to look

330
00:16:03,100 --> 00:16:03,879
pretty interesting.

331
00:16:03,879 --> 00:16:06,129
Let's substitute this expression with this one.

332
00:16:06,129 --> 00:16:07,700
We said that these are equivalent.

333
00:16:07,700 --> 00:16:10,000
And I'll switch colors, just for the sake of it.

334
00:16:10,000 --> 00:16:11,470
So we have the integral.

335
00:16:11,470 --> 00:16:13,250
From t is equal to a.

336
00:16:13,250 --> 00:16:17,809
Let me get our drawing back, if I-- from t is equal to a to t

337
00:16:17,809 --> 00:16:26,049
is equal to b of f of x of t times, or f of x of t and f of,

338
00:16:26,049 --> 00:16:29,809
or and y of t, they're both functions of t, and now instead

339
00:16:29,809 --> 00:16:34,519
of this expression, we can write the square root of, well,

340
00:16:34,519 --> 00:16:39,490
what's dx, what's the change in x with respect to, whatever

341
00:16:39,490 --> 00:16:40,149
this parameter is?

342
00:16:40,149 --> 00:16:41,699
What is dx dt?

343
00:16:41,700 --> 00:16:55,950
dx dt is the same thing as g prime of t.

344
00:16:55,950 --> 00:16:57,660
Right? x is a function of t.

345
00:16:57,659 --> 00:17:00,129
The function I wrote is g prime of t.

346
00:17:00,129 --> 00:17:06,400
And then dy dt is same thing as h prime of t.

347
00:17:06,400 --> 00:17:08,740
We could say that, you know, this function of t.

348
00:17:08,740 --> 00:17:09,690
So I just wanted to make that clear.

349
00:17:09,690 --> 00:17:11,759
We know these two functions, so we can just take their

350
00:17:11,759 --> 00:17:12,970
derivatives with respect to t.

351
00:17:12,970 --> 00:17:15,680
But I'm just going to leave it in that form.

352
00:17:15,680 --> 00:17:20,549
So the square root, and we take the derivative of x with

353
00:17:20,549 --> 00:17:26,029
respect to t squared, plus the derivative of y with respect to

354
00:17:26,029 --> 00:17:29,319
t squared, and all of that times dt.

355
00:17:29,319 --> 00:17:32,779
And this might look like some strange and convoluted formula,

356
00:17:32,779 --> 00:17:35,129
but this is actually something that we know how to deal with.

357
00:17:35,130 --> 00:17:39,340
We've now simplified this strange, you know, this

358
00:17:39,339 --> 00:17:42,649
arc-length problem, or this line integral, right?

359
00:17:42,650 --> 00:17:43,759
That's essentially what we're doing.

360
00:17:43,759 --> 00:17:46,759
We're taking an integral over a curve, or over a line,

361
00:17:46,759 --> 00:17:50,529
as opposed to just an interval on the x-axis.

362
00:17:50,529 --> 00:17:53,690
We've taken the strange line integral, that's in terms of

363
00:17:53,690 --> 00:17:56,730
the arc length of the line, and x's and y's, and we've put

364
00:17:56,730 --> 00:17:58,170
everything in terms of t.

365
00:17:58,170 --> 00:18:00,289
And I'm going to show you that in the next video, right?

366
00:18:00,289 --> 00:18:02,200
Everything is going to be expressed in terms of t,

367
00:18:02,200 --> 00:18:05,350
so this just turns into a simple, definite integral.

368
00:18:05,349 --> 00:18:07,029
So hopefully that didn't confuse you too much.

369
00:18:07,029 --> 00:18:09,079
I think you're going to see in the next video that this, right

370
00:18:09,079 --> 00:18:11,009
here, is actually a very straightforward

371
00:18:11,009 --> 00:18:11,849
thing to implement.

372
00:18:11,849 --> 00:18:14,250
And just to remind you where it all came from, I think I

373
00:18:14,250 --> 00:18:15,470
got the parentheses right.

374
00:18:15,470 --> 00:18:18,860
This right here was just a change in our arc length.

375
00:18:18,859 --> 00:18:20,389
That whole thing right there was just a

376
00:18:20,390 --> 00:18:21,330
change in arc length.

377
00:18:21,329 --> 00:18:26,369
And this is just the height of our function at that point.

378
00:18:26,369 --> 00:18:28,819
And we're just summing it, doing an infinite sum of

379
00:18:28,819 --> 00:18:30,369
infinitely small lengths.

380
00:18:30,369 --> 00:18:34,049
So this was a change in our arc length times the height.

381
00:18:34,049 --> 00:18:36,639
This is going to have an infinitely narrow width, and

382
00:18:36,640 --> 00:18:38,120
they're going to take an infinite number of these

383
00:18:38,119 --> 00:18:41,299
rectangles to get the area of this entire fence, or

384
00:18:41,299 --> 00:18:42,430
this entire curtain.

385
00:18:42,430 --> 00:18:45,210
And that's what this definite integral will give us,

386
00:18:45,210 --> 00:18:48,269
and we'll actually apply it in the next video.