1
00:00:00,000 --> 00:00:00,920


2
00:00:00,920 --> 00:00:03,819
In the last video, we had this rectangle, and we used a triple

3
00:00:03,819 --> 00:00:05,169
integral to figure out it's volume.

4
00:00:05,169 --> 00:00:08,000
And I know you were probably thinking, well, I could have

5
00:00:08,000 --> 00:00:12,099
just used my basic geometry to multiply the height times

6
00:00:12,099 --> 00:00:12,939
the width times the depth.

7
00:00:12,939 --> 00:00:15,719
And that's true because this was a constant-valued function.

8
00:00:15,720 --> 00:00:18,320
And then even once we evaluated, once we integrated

9
00:00:18,320 --> 00:00:20,629
with respect to z, we ended up with a double integral, which

10
00:00:20,629 --> 00:00:23,839
is exactly what you would have done in the last several videos

11
00:00:23,839 --> 00:00:26,579
when we just learned the volume under a surface.

12
00:00:26,579 --> 00:00:28,559
But then we added a twist at the end of the video.

13
00:00:28,559 --> 00:00:33,000
We said, fine, you could have figured out the volume within

14
00:00:33,000 --> 00:00:38,159
this rectangular domain, I guess, very straightforward

15
00:00:38,159 --> 00:00:39,000
using things you already knew.

16
00:00:39,000 --> 00:00:42,079
But what if our goal is not to figure out the volume?

17
00:00:42,079 --> 00:00:46,789
Our goal was to figure out the mass of this volume, and even

18
00:00:46,789 --> 00:00:50,240
more, the material that we're taking the volume of-- whether

19
00:00:50,240 --> 00:00:53,060
it's a volume of gas or a volume of some solid-- that

20
00:00:53,060 --> 00:00:55,050
its density is not constant.

21
00:00:55,049 --> 00:00:58,079
So now the mass becomes kind of-- I don't know--

22
00:00:58,079 --> 00:00:59,549
interesting to calculate.

23
00:00:59,549 --> 00:01:03,739
And so, what we defined, we defined a density function.

24
00:01:03,740 --> 00:01:07,769
And rho, this p looking thing with a curvy bottom-- that

25
00:01:07,769 --> 00:01:09,854
gives us the density at any given point.

26
00:01:09,855 --> 00:01:11,370
And at the end of the last video we said,

27
00:01:11,370 --> 00:01:12,829
well, what is mass?

28
00:01:12,829 --> 00:01:16,049
Mass is just density times volume.

29
00:01:16,049 --> 00:01:16,750
You could view it another way.

30
00:01:16,750 --> 00:01:21,170
Density is the same thing as mass divided by volume.

31
00:01:21,170 --> 00:01:26,629
So the mass around a very, very small point, and we called that

32
00:01:26,629 --> 00:01:29,750
d mass, the differential of the mass, is going equal the

33
00:01:29,750 --> 00:01:33,450
density at that point, or the rough density at exactly that

34
00:01:33,450 --> 00:01:36,790
point, times the volume differential around that point,

35
00:01:36,790 --> 00:01:40,100
times the volume of this little small cube.

36
00:01:40,099 --> 00:01:43,140
And then, as we saw it on the last video, if you're using

37
00:01:43,140 --> 00:01:46,239
rectangular coordinates, this volume differential could just

38
00:01:46,239 --> 00:01:50,390
be the x distance times the y distance times the z distance.

39
00:01:50,390 --> 00:01:55,689
So, the density was that our density function is defined

40
00:01:55,689 --> 00:01:57,730
to be x, y, and z, and we wanted to figure out the

41
00:01:57,730 --> 00:02:01,560
mass of this volume.

42
00:02:01,560 --> 00:02:04,140
And let's say that our x, y, and z coordinates-- their

43
00:02:04,140 --> 00:02:05,989
values, let's say they're in meters and let's say this

44
00:02:05,989 --> 00:02:09,340
density is in kilograms per meter cubed.

45
00:02:09,340 --> 00:02:12,270
So our answer is going to be in kilograms if that was the case.

46
00:02:12,270 --> 00:02:14,480
And those are kind of the traditional Si units.

47
00:02:14,479 --> 00:02:21,209
So let's figure out the mass of this variably dense volume.

48
00:02:21,210 --> 00:02:24,150
So all we do is we have the same integral up here.

49
00:02:24,150 --> 00:02:26,719


50
00:02:26,719 --> 00:02:29,859
So the differential of mass is going to be this value,

51
00:02:29,860 --> 00:02:30,996
so let's write that down.

52
00:02:30,996 --> 00:02:34,850


53
00:02:34,849 --> 00:02:38,919
It is x-- I want to make sure I don't run out of space.

54
00:02:38,919 --> 00:02:43,389
xyz times-- and I'm going to integrate with

55
00:02:43,389 --> 00:02:45,889
respect to dz first.

56
00:02:45,889 --> 00:02:47,909
But you could actually switch the order.

57
00:02:47,909 --> 00:02:49,750
Maybe we'll do that in the next video.

58
00:02:49,750 --> 00:02:55,810
We'll do dz first, then we'll do dy, then we'll do dx.

59
00:02:55,810 --> 00:03:00,120


60
00:03:00,120 --> 00:03:02,490
Once again, this is just the mass at any small

61
00:03:02,490 --> 00:03:04,310
differential of volume.

62
00:03:04,310 --> 00:03:07,759
And if we integrate with z first we said z goes from what?

63
00:03:07,759 --> 00:03:10,769
The boundaries on z were 0 to 2.

64
00:03:10,770 --> 00:03:14,050


65
00:03:14,050 --> 00:03:18,255
The boundaries on y were 0 to 4.

66
00:03:18,254 --> 00:03:21,109


67
00:03:21,110 --> 00:03:23,890
And the boundaries on x, x went from 0 to 3.

68
00:03:23,889 --> 00:03:26,750


69
00:03:26,750 --> 00:03:27,909
And how do we evaluate this?

70
00:03:27,909 --> 00:03:29,900
Well, what is the antiderivative-- we're

71
00:03:29,900 --> 00:03:31,370
integrating with respect to z first.

72
00:03:31,370 --> 00:03:35,659
So what's the antiderivative of xyz with respect to z?

73
00:03:35,659 --> 00:03:37,079
Well, let's see.

74
00:03:37,080 --> 00:03:45,080
This is just a constant so it'll be xyz squared over 2.

75
00:03:45,080 --> 00:03:46,040
Right?

76
00:03:46,039 --> 00:03:46,810
Yeah, that's right.

77
00:03:46,810 --> 00:03:52,689
And then we'll evaluate that from 2 to 0.

78
00:03:52,689 --> 00:03:54,870
And so you get-- I know I'm going to run out of space.

79
00:03:54,870 --> 00:03:59,420
So you're going to get 2 squared, which is 4,

80
00:03:59,419 --> 00:04:00,989
divided by 2, which is 2.

81
00:04:00,990 --> 00:04:05,460
So it's 2xy minus 0.

82
00:04:05,460 --> 00:04:09,070
So when you evaluate just this first we'll get 2xy, and

83
00:04:09,069 --> 00:04:11,409
now you have the other two integrals left.

84
00:04:11,409 --> 00:04:13,259
So I didn't write the other two integrals down.

85
00:04:13,259 --> 00:04:13,819
Maybe I'll write it down.

86
00:04:13,819 --> 00:04:16,680
So then you're left with two integrals.

87
00:04:16,680 --> 00:04:20,660
You're left with dy and dx.

88
00:04:20,660 --> 00:04:28,710
And y goes from 0 to 4 and x goes from 0 to 3.

89
00:04:28,709 --> 00:04:30,479
I'm definitely going to run out of space.

90
00:04:30,480 --> 00:04:32,200
And now you take the antiderivative of this

91
00:04:32,199 --> 00:04:34,110
with respect to y.

92
00:04:34,110 --> 00:04:36,639
So what's the antiderivative of this with respect to y?

93
00:04:36,639 --> 00:04:40,240
Let me erase some stuff just so I don't get too messy.

94
00:04:40,240 --> 00:04:44,230


95
00:04:44,230 --> 00:04:46,040
I was given the very good suggestion of making it

96
00:04:46,040 --> 00:04:48,340
scroll, but, unfortunately, I didn't make it scroll

97
00:04:48,339 --> 00:04:50,089
enough this time.

98
00:04:50,089 --> 00:04:54,159
So I can delete this stuff, I think.

99
00:04:54,160 --> 00:04:55,220
Oops, I deleted some of that.

100
00:04:55,220 --> 00:04:56,860
But you know what I deleted.

101
00:04:56,860 --> 00:04:58,290
OK, so let's take the antiderivative

102
00:04:58,290 --> 00:04:59,290
with respect to y.

103
00:04:59,290 --> 00:05:02,640
I'll start it up here where I have space.

104
00:05:02,639 --> 00:05:06,544
OK, so the antiderivative of 2xy with respect to y is y

105
00:05:06,545 --> 00:05:08,430
squared over 2, 2's cancel out.

106
00:05:08,430 --> 00:05:09,870
So you get xy squared.

107
00:05:09,870 --> 00:05:13,100


108
00:05:13,100 --> 00:05:15,270
And y goes from 0 to 4.

109
00:05:15,269 --> 00:05:18,000
And then we still have the outer integral to do.

110
00:05:18,000 --> 00:05:22,394
x goes from 0 to 3 dx.

111
00:05:22,394 --> 00:05:24,214
And when y is equal to 4 you get 16x.

112
00:05:24,214 --> 00:05:27,049


113
00:05:27,050 --> 00:05:29,050
And then when y is 0 the whole thing is 0.

114
00:05:29,050 --> 00:05:34,300
So you have 16x integrated from 0 to 3 dx.

115
00:05:34,300 --> 00:05:36,210
And that is equal to what?

116
00:05:36,209 --> 00:05:39,214
8x squared.

117
00:05:39,214 --> 00:05:42,699
And you evaluate it from 0 to 3.

118
00:05:42,699 --> 00:05:46,560
When it's 3, 8 times 9 is 72.

119
00:05:46,560 --> 00:05:49,040
And 0 times 8 is 0.

120
00:05:49,040 --> 00:05:51,810
So the mass of our figure-- the volume we figured out last

121
00:05:51,810 --> 00:05:53,230
time was 24 meters cubed.

122
00:05:53,230 --> 00:05:55,160
I erased it, but if you watched the last video

123
00:05:55,160 --> 00:05:56,210
that's what we learned.

124
00:05:56,209 --> 00:06:00,569
But it's mass is 72 kilograms.

125
00:06:00,569 --> 00:06:06,420
And we did that by integrating this 3 variable density

126
00:06:06,420 --> 00:06:08,090
function-- this function of 3 variables.

127
00:06:08,089 --> 00:06:10,229
Or in three-dimensions you can view it as a

128
00:06:10,230 --> 00:06:11,439
scalar field, right?

129
00:06:11,439 --> 00:06:13,910
At any given point, there is a value, but not

130
00:06:13,910 --> 00:06:14,420
really a direction.

131
00:06:14,420 --> 00:06:16,020
And that value is a density.

132
00:06:16,019 --> 00:06:20,539
But we integrated the scalar field in this volume.

133
00:06:20,540 --> 00:06:22,650
So that's kind of the new skill we learned with

134
00:06:22,649 --> 00:06:23,620
the triple integral.

135
00:06:23,620 --> 00:06:26,280
And in the next video I'll show you how to set up more

136
00:06:26,279 --> 00:06:27,459
complicated triple integrals.

137
00:06:27,459 --> 00:06:29,819
But the real difficulty with triple integrals is-- and I

138
00:06:29,819 --> 00:06:32,180
think you'll see that your calculus teacher will often do

139
00:06:32,180 --> 00:06:34,629
this-- when you're doing triple integrals, unless you have a

140
00:06:34,629 --> 00:06:38,290
very easy figure like this, the evaluation-- if you actually

141
00:06:38,290 --> 00:06:41,500
wanted to analytically evaluate a triple integral that has more

142
00:06:41,500 --> 00:06:44,910
complicated boundaries or more complicated for example,

143
00:06:44,910 --> 00:06:46,280
a density function.

144
00:06:46,279 --> 00:06:48,849
The integral gets very hairy, very fast.

145
00:06:48,850 --> 00:06:52,610
And it's often very difficult or very time consuming to

146
00:06:52,610 --> 00:06:55,759
evaluate it analytically just using your traditional

147
00:06:55,759 --> 00:06:56,269
calculus skills.

148
00:06:56,269 --> 00:06:59,789
So you'll see that on a lot of calculus exams when they start

149
00:06:59,790 --> 00:07:02,500
doing the triple integral, they just want you to set it up.

150
00:07:02,500 --> 00:07:05,519
They take your word for it that you've done so many integrals

151
00:07:05,519 --> 00:07:07,490
so far that you could take the antiderivative.

152
00:07:07,490 --> 00:07:09,819
And sometimes, if they really want to give you something more

153
00:07:09,819 --> 00:07:12,529
difficult they'll just say, well, switch the order.

154
00:07:12,529 --> 00:07:14,929
You know, this is the integral when we're dealing with

155
00:07:14,930 --> 00:07:16,699
respect to z, then y, then x.

156
00:07:16,699 --> 00:07:18,509
We want you to rewrite this integral when

157
00:07:18,509 --> 00:07:19,730
you switch the order.

158
00:07:19,730 --> 00:07:22,700
And we will do that in the next video.

159
00:07:22,699 --> 00:07:24,269
See you soon.

160
00:07:24,269 --> 00:07:25,500