1
00:00:00,000 --> 00:00:00,760


2
00:00:00,760 --> 00:00:04,389
In the last a video we figured out the volume between this

3
00:00:04,389 --> 00:00:08,570
surface, which was xy squared and the xy-plane when x went

4
00:00:08,570 --> 00:00:11,540
from 0 to 2 and y went from 0 to 1.

5
00:00:11,539 --> 00:00:13,389
And the way we did it is we integrated with

6
00:00:13,390 --> 00:00:15,890
respect to x first.

7
00:00:15,890 --> 00:00:20,679
We said, pick a y, and let's just figure out the

8
00:00:20,679 --> 00:00:21,859
area under the curve.

9
00:00:21,859 --> 00:00:24,640
And so we integrated with respect to x first, and then we

10
00:00:24,640 --> 00:00:25,850
integrated with respect to y.

11
00:00:25,850 --> 00:00:27,390
But we could have done it the other way around.

12
00:00:27,390 --> 00:00:31,080
So let's do that and just make sure we got the right answer.

13
00:00:31,079 --> 00:00:34,716
So let me erase a lot of this.

14
00:00:34,716 --> 00:00:37,636
So remember, our answer was 2/3 when we integrated with respect

15
00:00:37,636 --> 00:00:40,440
to x first, and then with respect to y.

16
00:00:40,439 --> 00:00:44,559
But I will show you that we can integrate the other way around.

17
00:00:44,560 --> 00:00:46,609
That's good when you can get the same answer in

18
00:00:46,609 --> 00:00:48,159
two different ways.

19
00:00:48,159 --> 00:00:52,399
So let me redraw that graph because I want to give

20
00:00:52,399 --> 00:00:54,570
you the intuition again.

21
00:00:54,570 --> 00:01:03,625
So that's my x-axis, y-axis, z-axis.

22
00:01:03,625 --> 00:01:10,849
x, y, z.

23
00:01:10,849 --> 00:01:13,539
Then this is my xy-plane.

24
00:01:13,540 --> 00:01:14,180
down here.

25
00:01:14,180 --> 00:01:16,990


26
00:01:16,989 --> 00:01:25,369
y goes from 0 to 1; x goes from 0 to 2, This is x equals 1,

27
00:01:25,370 --> 00:01:29,650
this is equals 2, this is y equals 1, and then the graph--

28
00:01:29,650 --> 00:01:33,290
I will do my best to draw it.

29
00:01:33,290 --> 00:01:39,060
Looks something-- let me get some contrast going here.

30
00:01:39,060 --> 00:01:41,310
So the graph looks something like this.

31
00:01:41,310 --> 00:01:45,379
Let me see if I can draw it.

32
00:01:45,379 --> 00:01:49,259
On this side it looks something like that and then it comes

33
00:01:49,260 --> 00:01:54,070
down like that, straight.

34
00:01:54,069 --> 00:01:55,909
And then, the volume we care about.

35
00:01:55,909 --> 00:01:57,924
It's actually this volume underneath the graph.

36
00:01:57,924 --> 00:02:01,679


37
00:02:01,680 --> 00:02:04,690
This is the top of the surface on that side.

38
00:02:04,689 --> 00:02:07,149
We care about this volume underneath the surface.

39
00:02:07,150 --> 00:02:09,730
And then when we draw the bottom of the surface-- let

40
00:02:09,729 --> 00:02:13,049
me do it in a darker color-- looks something like this.

41
00:02:13,050 --> 00:02:17,060
This is the bottom underneath the surface.

42
00:02:17,060 --> 00:02:19,060
I can even shade it a little bit just to show you that

43
00:02:19,060 --> 00:02:20,750
it's the underneath.

44
00:02:20,750 --> 00:02:24,009


45
00:02:24,009 --> 00:02:26,969
Hopefully that's a decent rendering of it.

46
00:02:26,969 --> 00:02:28,669
Let's look back at what we had before.

47
00:02:28,669 --> 00:02:30,839
It's like a page that I just flipped up at this point, and

48
00:02:30,840 --> 00:02:32,560
we care about this volume, kind of the colored

49
00:02:32,560 --> 00:02:35,120
area under there.

50
00:02:35,120 --> 00:02:36,250
So let's figure out how to do it.

51
00:02:36,250 --> 00:02:37,900
Last time we integrated with respect to x first.

52
00:02:37,900 --> 00:02:39,330
Let's integrate with respect to y first.

53
00:02:39,330 --> 00:02:41,520
So let's hold x constant.

54
00:02:41,520 --> 00:02:44,600
So if we hold x constant what we could do is for a given

55
00:02:44,599 --> 00:02:45,560
x-- let's pick an x.

56
00:02:45,560 --> 00:02:48,819


57
00:02:48,819 --> 00:02:55,169
So if we pick a given x, let's pick the x here.

58
00:02:55,169 --> 00:02:57,879


59
00:02:57,879 --> 00:03:01,900
Then what we can do, for a given x, you can view that

60
00:03:01,900 --> 00:03:03,310
function of x and y.

61
00:03:03,310 --> 00:03:07,069
If x is a constant, let's say if x is 1 then z is just

62
00:03:07,069 --> 00:03:08,489
equal to y squared.

63
00:03:08,490 --> 00:03:11,710
That's easy to figure out the area under because we can see

64
00:03:11,710 --> 00:03:13,650
that x isn't the constant, but we can treat it as a constant.

65
00:03:13,650 --> 00:03:16,640
So for example, for any given x, we would have

66
00:03:16,639 --> 00:03:17,729
a curve like this.

67
00:03:17,729 --> 00:03:22,560


68
00:03:22,560 --> 00:03:24,199
What we could do is we could try to figure out the

69
00:03:24,199 --> 00:03:27,609
area of this curve first.

70
00:03:27,610 --> 00:03:29,300
So how do we do that?

71
00:03:29,300 --> 00:03:32,650
Well, we just said, we could kind of view this function up

72
00:03:32,650 --> 00:03:35,860
here as z is equal to xy squared because that's

73
00:03:35,860 --> 00:03:36,830
exactly what it is.

74
00:03:36,830 --> 00:03:38,610
But we're holding x constant.

75
00:03:38,610 --> 00:03:41,010
We're treating it like a constant.

76
00:03:41,009 --> 00:03:44,359
To figure out that area we could take a dy, a change

77
00:03:44,360 --> 00:03:49,665
in y, multiply it by the height, which is xy squared.

78
00:03:49,664 --> 00:03:54,129


79
00:03:54,129 --> 00:04:02,829
So we take xy squared, multiply it by dy, and then if we want

80
00:04:02,830 --> 00:04:05,020
this entire area we integrate it from y is equal to

81
00:04:05,020 --> 00:04:06,909
0 to y is equal to 1.

82
00:04:06,909 --> 00:04:11,079


83
00:04:11,080 --> 00:04:11,600
Fair enough.

84
00:04:11,599 --> 00:04:14,569
Now once we have that area, if you want the volume underneath

85
00:04:14,569 --> 00:04:18,209
this entire surface what we could do is we can multiply

86
00:04:18,209 --> 00:04:22,449
this area times dx and get some depth going.

87
00:04:22,449 --> 00:04:24,779
Let me pick a nice color, that's green.

88
00:04:24,779 --> 00:04:27,119
So that's our dx.

89
00:04:27,120 --> 00:04:29,639
So if we multiply that times dx we would get some depth.

90
00:04:29,639 --> 00:04:32,949
Let me do a darker color, get some contrast.

91
00:04:32,949 --> 00:04:37,079
Sometimes I feel like that guy who paints on PBS.

92
00:04:37,079 --> 00:04:39,819
So now we have the volume of this, you kind of view it-- the

93
00:04:39,819 --> 00:04:42,949
area under the curve times a dx, so we have some depth here.

94
00:04:42,949 --> 00:04:44,750
So it's time dx.

95
00:04:44,750 --> 00:04:47,540
And if we want to figure out the entire volume under this

96
00:04:47,540 --> 00:04:51,030
surface-- between the surface and the xy-plane given this

97
00:04:51,029 --> 00:04:55,369
constraint to our domain-- we just integrate from x

98
00:04:55,370 --> 00:04:56,930
is equal to 0 to 2.

99
00:04:56,930 --> 00:05:02,120


100
00:05:02,120 --> 00:05:05,050
All right, so let's think about it.

101
00:05:05,050 --> 00:05:07,870
This area in green here that we started with, that

102
00:05:07,870 --> 00:05:09,100
should be a function of x.

103
00:05:09,100 --> 00:05:11,600
We held x constant, but depending on which x you pick

104
00:05:11,600 --> 00:05:13,780
this area is going to change.

105
00:05:13,779 --> 00:05:16,819
So when we evaluate this magenta inner integral with

106
00:05:16,819 --> 00:05:19,329
respect to y we should get a function of x.

107
00:05:19,329 --> 00:05:20,469
And then when you evaluate the whole thing we'll

108
00:05:20,470 --> 00:05:21,220
get our volumes.

109
00:05:21,220 --> 00:05:22,320
So let's do it.

110
00:05:22,319 --> 00:05:24,699
Let's evaluate this inner integral.

111
00:05:24,699 --> 00:05:26,420
Hold x constant.

112
00:05:26,420 --> 00:05:28,090
What's the antiderivative of y squared?

113
00:05:28,089 --> 00:05:30,019
It's y of the third over 3.

114
00:05:30,019 --> 00:05:35,529


115
00:05:35,529 --> 00:05:37,585
The x is a constant, right?

116
00:05:37,586 --> 00:05:42,290
We're going to evaluate that at 1 and at 0.

117
00:05:42,290 --> 00:05:47,670
The outer integral is still with respect to x dx.

118
00:05:47,670 --> 00:05:49,730
This is equal to-- let's see.

119
00:05:49,730 --> 00:05:52,860
When you evaluate y is equal to 1 you get 1 to the third.

120
00:05:52,860 --> 00:05:53,370
That's 1.

121
00:05:53,370 --> 00:05:58,050
So it's x/3 minus when y is 0 then that whole

122
00:05:58,050 --> 00:06:00,560
thing just becomes 0.

123
00:06:00,560 --> 00:06:02,870
This purple expression is just x/3.

124
00:06:02,870 --> 00:06:06,350


125
00:06:06,350 --> 00:06:11,120
And then we still have the outside integral

126
00:06:11,120 --> 00:06:13,560
from 0 to 2 dx.

127
00:06:13,560 --> 00:06:18,290
So given what x we have, the area of this green surface--

128
00:06:18,290 --> 00:06:20,410
that was where we started.

129
00:06:20,410 --> 00:06:25,430
Given any given x, that area-- I wanted something

130
00:06:25,430 --> 00:06:26,699
with some contrast.

131
00:06:26,699 --> 00:06:32,550
This area is x/3 depending on which x you pick.

132
00:06:32,550 --> 00:06:35,829
If x is 1, this area right here is 1/3.

133
00:06:35,829 --> 00:06:38,579
But now we're going to integrate underneath the entire

134
00:06:38,579 --> 00:06:40,550
surface and get our volume.

135
00:06:40,550 --> 00:06:42,160
And like I said, when you integrate it,

136
00:06:42,160 --> 00:06:43,870
it's a function of x.

137
00:06:43,870 --> 00:06:45,220
So let's do that.

138
00:06:45,220 --> 00:06:51,440
And this is just plain old vanilla, standard integral.

139
00:06:51,439 --> 00:06:52,930
So what's the antiderivative of x?

140
00:06:52,930 --> 00:06:54,930
It's x squared over 2.

141
00:06:54,930 --> 00:06:58,889
We have a 1/3 there so it equals x squared

142
00:06:58,889 --> 00:07:01,180
over 2 times 3.

143
00:07:01,180 --> 00:07:04,410
So x squared over 6.

144
00:07:04,410 --> 00:07:07,410
And we're going to evaluate it at 2 and at 0.

145
00:07:07,410 --> 00:07:11,030
2 squared over 6 is 4/6.

146
00:07:11,029 --> 00:07:14,109
Minus 0/6, which is equal to 0.

147
00:07:14,110 --> 00:07:15,639
Equals 4/6.

148
00:07:15,639 --> 00:07:16,560
What is 4/6?

149
00:07:16,560 --> 00:07:19,850
Well, that's just the same thing as 2/3.

150
00:07:19,850 --> 00:07:23,360
So the volume under the surface is 2/3, and if you watched

151
00:07:23,360 --> 00:07:29,150
the previous video you will appreciate the fact that when

152
00:07:29,149 --> 00:07:32,589
we integrated the other way around, when we did it with

153
00:07:32,589 --> 00:07:35,539
respect to x first and then y, we got the exact same answer.

154
00:07:35,540 --> 00:07:38,950
So the universe is in proper working order.

155
00:07:38,949 --> 00:07:41,579
And I've surprisingly, actually finished this

156
00:07:41,579 --> 00:07:43,089
video with extra time.

157
00:07:43,089 --> 00:07:47,399
So for fun, we can just spin this graph and just appreciate

158
00:07:47,399 --> 00:07:52,289
the fact that we have figured out the volume between this

159
00:07:52,290 --> 00:07:56,710
surface, xy squared and the xy-plane.

160
00:07:56,709 --> 00:07:58,949
Pretty neat.

161
00:07:58,949 --> 00:08:02,360
Anyway, I'll see you in the next video.

162
00:08:02,360 --> 00:08:03,000