1
00:00:00,000 --> 00:00:00,570


2
00:00:00,570 --> 00:00:05,480
So let's say I have a function of x and y; f of x and y is

3
00:00:05,480 --> 00:00:08,470
equal to x plus y squared.

4
00:00:08,470 --> 00:00:11,400
If I try to draw that, let's see if I can have

5
00:00:11,400 --> 00:00:13,099
a good attempt at it.

6
00:00:13,099 --> 00:00:16,589
That is my y axis-- I'm going to do a little perspective here

7
00:00:16,589 --> 00:00:21,439
--this is my x axis-- I make do the negative x and y axis,

8
00:00:21,440 --> 00:00:24,179
could do it in that direction --this is my x axis here.

9
00:00:24,179 --> 00:00:27,149
And if I were to graph this when y is 0, it's going to be

10
00:00:27,149 --> 00:00:29,719
just a-- let me draw it in yellow --is going to be just a

11
00:00:29,719 --> 00:00:32,310
straight line that looks something like that.

12
00:00:32,310 --> 00:00:34,050
And then for any given actually, we're going to

13
00:00:34,049 --> 00:00:36,009
have a parabola in y.

14
00:00:36,009 --> 00:00:39,229
y is going to look something like that.

15
00:00:39,229 --> 00:00:40,334
I'm just going to it in the positive quadrant.

16
00:00:40,335 --> 00:00:41,780
It's going to look something like that.

17
00:00:41,780 --> 00:00:44,980


18
00:00:44,979 --> 00:00:47,089
It'll actually, when you go into the negative y, you're

19
00:00:47,090 --> 00:00:48,905
going to see the other half of the parabola, but I'm not going

20
00:00:48,905 --> 00:00:50,370
to worry about it too much.

21
00:00:50,369 --> 00:00:51,979
So you're going to have this surface.

22
00:00:51,979 --> 00:00:53,189
it looks something like that.

23
00:00:53,189 --> 00:00:55,140
Maybe I'll do to another attempt at drawing it.

24
00:00:55,140 --> 00:00:57,929
But this is our ceiling we're going to deal with again.

25
00:00:57,929 --> 00:01:03,340
And then I'm going to have a path in the xy plane.

26
00:01:03,340 --> 00:01:08,659
I'm going to start at the point 2 comma 0. x is

27
00:01:08,659 --> 00:01:10,429
equal to 2, y is 0.

28
00:01:10,430 --> 00:01:13,670
And I'm going to travel, just like we did in the last video,

29
00:01:13,670 --> 00:01:16,379
I'm going to travel along a circle, but this time the

30
00:01:16,379 --> 00:01:18,629
circle's going to have of radius 2.

31
00:01:18,629 --> 00:01:20,649
Move counter clockwise in that circle.

32
00:01:20,650 --> 00:01:22,609
This is on the xy plane, just to be able to

33
00:01:22,609 --> 00:01:24,040
visualize it properly.

34
00:01:24,040 --> 00:01:26,500
So this right here's a point 0, 2.

35
00:01:26,500 --> 00:01:30,055
And I'm going to come back along the y axis.

36
00:01:30,055 --> 00:01:35,640
This is my path; I'm going to come back along the y access

37
00:01:35,640 --> 00:01:38,219
and then so I look a left here, and then I'm going to take

38
00:01:38,219 --> 00:01:40,640
another left here in and come back along the x axis.

39
00:01:40,640 --> 00:01:44,930


40
00:01:44,930 --> 00:01:46,540
I drew it in these two shades of green.

41
00:01:46,540 --> 00:01:48,680
That is my contour.

42
00:01:48,680 --> 00:01:52,830
And what I want to do is I want to evaluate the surface area

43
00:01:52,829 --> 00:01:56,260
of essentially this little building that has the roof of f

44
00:01:56,260 --> 00:02:00,020
of xy is equal to x plus y squared, and I want to find the

45
00:02:00,019 --> 00:02:01,619
surface area of its walls.

46
00:02:01,620 --> 00:02:05,704
So you'll have this wall right here, whose base is the x axis.

47
00:02:05,704 --> 00:02:09,250
Then you're going to have this wall, which is along the curve;

48
00:02:09,250 --> 00:02:14,490
it's going to look something like kind of funky wall on

49
00:02:14,490 --> 00:02:16,409
that curved side right there.

50
00:02:16,409 --> 00:02:19,549
I'll try my best effort to try to-- it's going to be

51
00:02:19,550 --> 00:02:25,510
curving way up like that and then along the y axis.

52
00:02:25,509 --> 00:02:28,269


53
00:02:28,270 --> 00:02:32,240
It's going to have like a half a parabolic wall right there.

54
00:02:32,240 --> 00:02:34,150
I'll do that back wall along the y axis.

55
00:02:34,150 --> 00:02:38,400
I'll do that in orange, I'll use magenta.

56
00:02:38,400 --> 00:02:41,530
That is the back wall along the y axis.

57
00:02:41,530 --> 00:02:45,659
Then you have this front wall along the x axis.

58
00:02:45,659 --> 00:02:50,229
And then you have this weird curvy curtain or wall-- do that

59
00:02:50,229 --> 00:02:55,019
maybe in blue --that goes along this curve right here, this

60
00:02:55,020 --> 00:02:56,740
part of a circle of radius 2.

61
00:02:56,740 --> 00:02:58,890
So hopefully you get that visualization.

62
00:02:58,889 --> 00:03:00,669
It's a little harder; I'm not using any graphic

63
00:03:00,669 --> 00:03:01,679
program at this time.

64
00:03:01,680 --> 00:03:03,900
But I want to figure out the surface area, the

65
00:03:03,900 --> 00:03:06,270
combined surface area of these three walls.

66
00:03:06,270 --> 00:03:09,390
And in very simple notation we could say, well, the surface

67
00:03:09,389 --> 00:03:16,429
area of those walls-- of this wall plus that wall plus that

68
00:03:16,430 --> 00:03:22,569
wall --is going to be equal to the line integral along this

69
00:03:22,569 --> 00:03:26,019
curve, or along this contour-- however you want to call it

70
00:03:26,020 --> 00:03:32,820
--of f of xy,-- so that's x plus y squared --ds, where ds

71
00:03:32,819 --> 00:03:35,219
is just a little length along our contour.

72
00:03:35,219 --> 00:03:37,449
And since this is a closed loop, we'll call this a

73
00:03:37,449 --> 00:03:38,500
closed line interval.

74
00:03:38,500 --> 00:03:40,560
And we'll sometimes see this notation right here.

75
00:03:40,560 --> 00:03:43,080


76
00:03:43,080 --> 00:03:44,480
Often you'll see that in physics books.

77
00:03:44,479 --> 00:03:45,869
And we'll be dealing with a lot more.

78
00:03:45,870 --> 00:03:47,330
And we'll put a circle on the interval sign.

79
00:03:47,330 --> 00:03:50,060
And all that means is that the contour we're dealing with is a

80
00:03:50,060 --> 00:03:53,750
closed contour; we get back to where we started from.

81
00:03:53,750 --> 00:03:55,770
But how do we solve this thing?

82
00:03:55,770 --> 00:03:57,659
A good place to start is to just to find

83
00:03:57,659 --> 00:03:58,629
the contour itself.

84
00:03:58,629 --> 00:04:00,829
And just to simply it, we're going to divide it into three

85
00:04:00,830 --> 00:04:03,040
pieces and it essentially just do three separate

86
00:04:03,039 --> 00:04:04,289
line integrals.

87
00:04:04,289 --> 00:04:08,769
Because you know, this isn't a very continuous contour.

88
00:04:08,770 --> 00:04:10,050
so the first part.

89
00:04:10,050 --> 00:04:11,980
Let's do this first part of the curve where we're going

90
00:04:11,979 --> 00:04:15,639
along a circle of radius 2.

91
00:04:15,639 --> 00:04:20,769
And that's pretty easy to construct if we have x-- let me

92
00:04:20,769 --> 00:04:25,849
do each part of the contour in a different color, so if I do

93
00:04:25,850 --> 00:04:30,920
orange this part of the contour --if we say that x is equal 2

94
00:04:30,920 --> 00:04:39,439
cosine of t and y is equal to 2 sine of t and if we say that

95
00:04:39,439 --> 00:04:42,680
t-- and this is really just building off what we saw on the

96
00:04:42,680 --> 00:04:46,959
last video --if we say that t-- and that this is from t is a

97
00:04:46,959 --> 00:04:54,169
greater than or equal to 0 and is less than or equal to pi

98
00:04:54,170 --> 00:04:58,270
over 2 --t is essentially going to be the angle that

99
00:04:58,269 --> 00:05:00,199
we're going along the circle right here.

100
00:05:00,199 --> 00:05:02,039
This will actually describe this path.

101
00:05:02,040 --> 00:05:03,819
And if you know, how I constructed this is little

102
00:05:03,819 --> 00:05:05,980
confusing, you might want to review the video on

103
00:05:05,980 --> 00:05:07,860
parametric equations.

104
00:05:07,860 --> 00:05:09,360
So this is the first part of our path.

105
00:05:09,360 --> 00:05:14,080
So if we just wanted to find the surface area of that wall

106
00:05:14,079 --> 00:05:15,939
right there, we know we're going to have to find

107
00:05:15,939 --> 00:05:17,670
dx, dt and dy, dt.

108
00:05:17,670 --> 00:05:19,970
So let's get that out of the way right now.

109
00:05:19,970 --> 00:05:28,770
So if we say dx, dt is going to be equal to minus 2, sine of t,

110
00:05:28,769 --> 00:05:34,495
dy, dy is going to be equal to 2 cosine of t; just the

111
00:05:34,495 --> 00:05:35,840
derivatives of these.

112
00:05:35,839 --> 00:05:37,349
We've seen that many times before.

113
00:05:37,350 --> 00:05:41,290
So it we want this orange wall's surface area, we can

114
00:05:41,290 --> 00:05:43,675
take the integral-- and if any of this is confusing, there are

115
00:05:43,675 --> 00:05:47,400
two videos before this where we kind of derive this formula

116
00:05:47,399 --> 00:05:51,169
--but we could take the integral from t is equal to 0

117
00:05:51,170 --> 00:05:59,259
to pi over 2 our function of x plus y squared and

118
00:05:59,259 --> 00:06:01,610
then times the ds.

119
00:06:01,610 --> 00:06:03,569
So x plus y squared will give the height of

120
00:06:03,569 --> 00:06:04,550
each little block.

121
00:06:04,550 --> 00:06:06,360
And then we want to get the width of each little block,

122
00:06:06,360 --> 00:06:10,689
which is ds, but we know that we can rewrite the ds as the

123
00:06:10,689 --> 00:06:18,930
square root-- give myself some room right here --of dx of the

124
00:06:18,930 --> 00:06:24,310
derivative of x with respect to t squared-- so that is minus 2

125
00:06:24,310 --> 00:06:29,160
sine of t squared --plus the derivative of y with

126
00:06:29,160 --> 00:06:36,660
respect to t squared, dt.

127
00:06:36,660 --> 00:06:39,050
This will give us the orange section, and then we can worry

128
00:06:39,050 --> 00:06:41,819
about the other two walls.

129
00:06:41,819 --> 00:06:43,409
And so how can we simplify this?

130
00:06:43,410 --> 00:06:49,670
Well, this is going to be equal to the integral from 0 to pi

131
00:06:49,670 --> 00:06:55,000
over 2 of x plus y squared.

132
00:06:55,000 --> 00:06:59,180
And actually, let me write everything in terms of t.

133
00:06:59,180 --> 00:07:01,250
So x is equal to 2 cosine of t.

134
00:07:01,250 --> 00:07:02,910
So let me write that down.

135
00:07:02,910 --> 00:07:13,310
So it's 2 cosine of t plus y, which is 2 sine of t, and we're

136
00:07:13,310 --> 00:07:14,829
going to square everything.

137
00:07:14,829 --> 00:07:18,779
And then all of that times this crazy radical.

138
00:07:18,779 --> 00:07:22,879
Right now it looks like a hard antiderivative or integral to

139
00:07:22,879 --> 00:07:24,779
solve, but I we'll find out it's not too bad.

140
00:07:24,779 --> 00:07:30,379
This is going to be equal to 4 sine squared of t plus

141
00:07:30,379 --> 00:07:34,379
4 cosine squared of t.

142
00:07:34,379 --> 00:07:37,019
We can factor a 4 out.

143
00:07:37,019 --> 00:07:39,449
I don't want to forget the dt.

144
00:07:39,449 --> 00:07:41,819
This over here-- let me just simplify this expression so I

145
00:07:41,819 --> 00:07:43,259
don't have to keep rewriting it.

146
00:07:43,259 --> 00:07:47,819
That is the same thing is the square root of 4 times

147
00:07:47,819 --> 00:07:52,649
sine squared of t plus cosine squared of t.

148
00:07:52,649 --> 00:07:54,799
We know what that is: that's just 1.

149
00:07:54,800 --> 00:07:56,516
So this whole thing just simplifies to the square

150
00:07:56,516 --> 00:07:58,650
root of 4, which is just 2.

151
00:07:58,649 --> 00:08:02,120
So this whole thing simplifies to 2, which is nice for

152
00:08:02,120 --> 00:08:04,290
solving our antiderivative.

153
00:08:04,290 --> 00:08:05,860
That means simplifying things a lot.

154
00:08:05,860 --> 00:08:09,620
So this whole thing simplifies down to-- I'll do it over here.

155
00:08:09,620 --> 00:08:12,319
I don't want to waste too much space; I have two more walls to

156
00:08:12,319 --> 00:08:16,759
figure out --the integral from t is equal to 0 to pi over 2.

157
00:08:16,759 --> 00:08:17,659
I want to make it very clear.

158
00:08:17,660 --> 00:08:20,700
I just chose the simplest parametrization I

159
00:08:20,699 --> 00:08:22,079
could for x and y.

160
00:08:22,079 --> 00:08:23,629
But I could have picked other parametrizations, but

161
00:08:23,629 --> 00:08:25,250
then I would have had to change t accordingly.

162
00:08:25,250 --> 00:08:28,199
So as long as you're consistent with how you do it, it

163
00:08:28,199 --> 00:08:29,019
should all work out.

164
00:08:29,019 --> 00:08:32,289
There isn't just one parametrization for this curve;

165
00:08:32,289 --> 00:08:34,129
it's kind of depending on how fast you want to go

166
00:08:34,129 --> 00:08:35,139
along the curve.

167
00:08:35,139 --> 00:08:39,449
Watch the parametric functions videos if you want a little

168
00:08:39,450 --> 00:08:40,900
bit more depth on that.

169
00:08:40,899 --> 00:08:41,860
Anyway, this thing simplifies.

170
00:08:41,860 --> 00:08:44,399
We have a 2 here; 2 times cosine of t, that's

171
00:08:44,399 --> 00:08:47,720
4 cosine of t.

172
00:08:47,720 --> 00:08:52,500
And then here we have 2 sine squared sine of t squared.

173
00:08:52,500 --> 00:08:54,429
So that's 4 sine squared of t.

174
00:08:54,429 --> 00:08:58,389


175
00:08:58,389 --> 00:09:01,230
And then we have to multiply times this 2 again, so

176
00:09:01,230 --> 00:09:02,970
that gives us an 8.

177
00:09:02,970 --> 00:09:07,750
8 time sine squared of t, dt.

178
00:09:07,750 --> 00:09:10,039
And then you know, sine squared of t; that looks like a

179
00:09:10,039 --> 00:09:12,839
tough thing to find the antiderivative for, but we can

180
00:09:12,840 --> 00:09:18,149
remember that sine squared of, really anything-- we could say

181
00:09:18,149 --> 00:09:21,659
sine squared of u is equal to 1/2 half times 1

182
00:09:21,659 --> 00:09:24,289
minus cosine of 2u.

183
00:09:24,289 --> 00:09:26,699
So we can reuse this identity.

184
00:09:26,700 --> 00:09:30,379
I can try the t here; sine squared of t is equal to 1/2

185
00:09:30,379 --> 00:09:33,539
times 1 minus cosine of 2t.

186
00:09:33,539 --> 00:09:35,049
Let me rewrite it that way because that'll make the

187
00:09:35,049 --> 00:09:36,389
integral a lot easier to solve.

188
00:09:36,389 --> 00:09:40,529
So we get integral from 0 the pi over 2-- and actually I

189
00:09:40,529 --> 00:09:45,079
could break up, well I won't break it up --of 4 cosine of

190
00:09:45,080 --> 00:09:51,550
t plus 8 times this thing.

191
00:09:51,549 --> 00:09:53,349
8 times this thing; this is the same thing as

192
00:09:53,350 --> 00:09:55,019
sine squared of t.

193
00:09:55,019 --> 00:10:02,250
So 8 times this-- 8 times 1/2 is 4 --4 times 1 minus cosine

194
00:10:02,250 --> 00:10:05,590
of 2t-- just use a little trig identity there --and

195
00:10:05,590 --> 00:10:07,389
all of that dt.

196
00:10:07,389 --> 00:10:09,429
Now this should be reasonably straight forward to get

197
00:10:09,429 --> 00:10:10,799
the antiderivative of.

198
00:10:10,799 --> 00:10:11,979
Let's just take it.

199
00:10:11,980 --> 00:10:16,500
The antiderivative of this is antiderivative of cosine

200
00:10:16,500 --> 00:10:18,610
of t; that's a sine of t.

201
00:10:18,610 --> 00:10:20,149
The derivative of sine is cosine.

202
00:10:20,149 --> 00:10:25,049
So this is going to be 4 sine of t-- the scalars don't affect

203
00:10:25,049 --> 00:10:28,469
anything --and then, well let me just distribute this 4.

204
00:10:28,470 --> 00:10:34,639
So this is 4 times 1 which is 4 minus 4 cosine of 2t.

205
00:10:34,639 --> 00:10:39,629
So the antiderivative of 4 is 4t-- plus 4t --and then the

206
00:10:39,629 --> 00:10:43,879
antiderivative of minus 4 cosine of u00b5 t?

207
00:10:43,879 --> 00:10:45,970
Let's see it's going to be sine of 2t.

208
00:10:45,970 --> 00:10:51,590


209
00:10:51,590 --> 00:10:58,899
The derivative of sine of 2t is 2 cosine of 2t.

210
00:10:58,899 --> 00:11:01,659
We're going to have to have a minus sign there, and put a 2

211
00:11:01,659 --> 00:11:03,419
there, and now it should work out.

212
00:11:03,419 --> 00:11:05,740
What's the derivative of minus 2 sine of t?

213
00:11:05,740 --> 00:11:07,240
Take the derivative of the inside 2 times

214
00:11:07,240 --> 00:11:09,399
minus 2 is minus 4.

215
00:11:09,399 --> 00:11:11,740
And the derivative of sine of 2t with respect to

216
00:11:11,740 --> 00:11:13,279
2t is cosine of 2t.

217
00:11:13,279 --> 00:11:15,829
So there we go; we've figured out our antiderivative.

218
00:11:15,830 --> 00:11:19,259
Now we evaluate it from 0 the pi over 2.

219
00:11:19,259 --> 00:11:22,000


220
00:11:22,000 --> 00:11:23,419
And what do we get?

221
00:11:23,419 --> 00:11:27,209
We get 4 sine-- let me write this down, for I don't want to

222
00:11:27,210 --> 00:11:34,100
skip too many --sine of pi over 2 plus 4 times pi over 2--

223
00:11:34,100 --> 00:11:42,040
that's just 2 pi minus 2 sine of 2 times pi over 2 sine of

224
00:11:42,039 --> 00:11:48,620
pie, and then all of that minus all this evaluated at 0.

225
00:11:48,620 --> 00:11:50,580
That's actually pretty straightforward because

226
00:11:50,580 --> 00:11:52,670
sine of 0 is 0.

227
00:11:52,669 --> 00:11:56,490
4 times 0 is 0, and sine of 2 times 0, that's also 0.

228
00:11:56,490 --> 00:11:59,190
So everything with the 0's work out nicely.

229
00:11:59,190 --> 00:12:00,350
And then what do we have here?

230
00:12:00,350 --> 00:12:05,769
Sine of pi over 2-- in my head, I think sine of 90 degrees;

231
00:12:05,769 --> 00:12:09,319
same thing --that is 1.

232
00:12:09,320 --> 00:12:11,980
And then sine of pi is 0, that's 180 degrees.

233
00:12:11,980 --> 00:12:13,670
So this whole thing cancels out.

234
00:12:13,669 --> 00:12:17,439
So we're left with 4 plus 2 pi.

235
00:12:17,440 --> 00:12:23,290
So just like that we were able to figure out the area of this

236
00:12:23,289 --> 00:12:25,519
first curvy wall here, and frankly, that's

237
00:12:25,519 --> 00:12:26,710
the hardest part.

238
00:12:26,710 --> 00:12:29,759
Now let's figure out the area of this curve.

239
00:12:29,759 --> 00:12:31,220
And actually you're going to find out that these other

240
00:12:31,220 --> 00:12:33,830
curves as they go along the axes are much, much, much

241
00:12:33,830 --> 00:12:35,759
easier, but we're going to have to find different

242
00:12:35,759 --> 00:12:37,480
parametrizations for this.

243
00:12:37,480 --> 00:12:43,389
So if we take this curve right here, let's do a

244
00:12:43,389 --> 00:12:44,590
parametrization for that.

245
00:12:44,590 --> 00:12:45,490
Actually, you know what?

246
00:12:45,490 --> 00:12:48,629
Let me continue this in the next video because I realize

247
00:12:48,629 --> 00:12:49,919
I've been running a little longer.

248
00:12:49,919 --> 00:12:53,110
I'll do the next two walls and then we'll sum them all up.