WEBVTT
1
00:00:01.500 --> 00:00:06.379 A:middle L:90%
given this integral 01 and then the aero from 3
2
00:00:06.379 --> 00:00:10.419 A:middle L:90%
to 6 of the function X times the square root
3
00:00:10.429 --> 00:00:14.740 A:middle L:90%
of X squared plus three wine dx dy y.
4
00:00:15.240 --> 00:00:18.690 A:middle L:90%
We want to find a numerical answer for this double
5
00:00:18.690 --> 00:00:23.969 A:middle L:90%
integral. Okay, so let's start off by evaluating
6
00:00:23.980 --> 00:00:27.339 A:middle L:90%
this inside Integral first. So we have the integral
7
00:00:27.339 --> 00:00:31.149 A:middle L:90%
from 3 to 6 of the function x times X
8
00:00:31.160 --> 00:00:33.850 A:middle L:90%
where the square root of X squared plus three y
9
00:00:34.840 --> 00:00:40.619 A:middle L:90%
dx Um So what we're gonna dio is we're going
10
00:00:40.619 --> 00:00:42.829 A:middle L:90%
to use U substitution. So we're gonna let u
11
00:00:42.829 --> 00:00:49.399 A:middle L:90%
equal basically X squared plus three y And then since
12
00:00:49.399 --> 00:00:53.560 A:middle L:90%
we're integrating respect to DX, um, basically,
13
00:00:53.560 --> 00:00:57.429 A:middle L:90%
this use substitution will be for X, So we're
14
00:00:57.429 --> 00:01:02.899 A:middle L:90%
gonna be driving you or the left side by you
15
00:01:04.739 --> 00:01:07.510 A:middle L:90%
So the left side will be to you, and
16
00:01:07.510 --> 00:01:08.540 A:middle L:90%
then we're gonna derive the right side by X.
17
00:01:10.140 --> 00:01:14.159 A:middle L:90%
So when we drive the right side by X,
18
00:01:14.170 --> 00:01:18.549 A:middle L:90%
we consider every other variable to be constant. So
19
00:01:18.969 --> 00:01:19.780 A:middle L:90%
we have to acts, obviously. But then this
20
00:01:19.780 --> 00:01:22.750 A:middle L:90%
three, why could be considered as a constant?
21
00:01:23.239 --> 00:01:26.540 A:middle L:90%
So this just just result in zero Andhra member to
22
00:01:26.540 --> 00:01:34.780 A:middle L:90%
multiply by d accept and so And then what I
23
00:01:34.780 --> 00:01:38.359 A:middle L:90%
like to do is I want this to match up
24
00:01:38.370 --> 00:01:42.459 A:middle L:90%
with what I have in our integral. So I
25
00:01:42.459 --> 00:01:45.930 A:middle L:90%
just divide to buy, but on both sides and
26
00:01:45.930 --> 00:01:49.689 A:middle L:90%
I get that x d x is equal to do
27
00:01:49.689 --> 00:01:55.349 A:middle L:90%
you over to Okay, so, plugging all this
28
00:01:55.349 --> 00:02:00.840 A:middle L:90%
back in we have that, um, are new
29
00:02:00.840 --> 00:02:06.180 A:middle L:90%
. Integral is, um basically, I should so
30
00:02:06.180 --> 00:02:08.449 A:middle L:90%
, too with some stew x dx with do you
31
00:02:08.449 --> 00:02:13.500 A:middle L:90%
over to so 1/2 and then do you I'm just
32
00:02:14.060 --> 00:02:15.800 A:middle L:90%
taking out the constant right here. Just make it
33
00:02:15.800 --> 00:02:21.629 A:middle L:90%
easier. And then this square root of X squared
34
00:02:21.629 --> 00:02:23.759 A:middle L:90%
plus three, why should be just to live square
35
00:02:23.759 --> 00:02:27.689 A:middle L:90%
of you now looking at the limits. Um,
36
00:02:28.310 --> 00:02:30.930 A:middle L:90%
since we're deriving respect to you now we have to
37
00:02:30.930 --> 00:02:37.250 A:middle L:90%
find the U Uh, basically, you limits instead
38
00:02:37.840 --> 00:02:42.030 A:middle L:90%
. So we're gonna plug in our ex limits and
39
00:02:42.030 --> 00:02:43.870 A:middle L:90%
sue here in order to get Are you limits?
40
00:02:44.539 --> 00:02:49.259 A:middle L:90%
So playing three for X? Um, we have
41
00:02:49.340 --> 00:02:54.860 A:middle L:90%
that you is equal to nine plus three y and
42
00:02:54.860 --> 00:03:00.689 A:middle L:90%
then plaguing six and two here. We should get
43
00:03:00.139 --> 00:03:05.080 A:middle L:90%
36 plus three Y Okay, So this is our
44
00:03:05.080 --> 00:03:08.590 A:middle L:90%
new integral. Um, Now, evaluating this can
45
00:03:08.590 --> 00:03:10.210 A:middle L:90%
get a little messy, but bear with me.
46
00:03:10.400 --> 00:03:16.879 A:middle L:90%
Um, we have 1/2 times we're gonna use the
47
00:03:16.889 --> 00:03:20.900 A:middle L:90%
reverse spiral again. So we're gonna add one to
48
00:03:20.900 --> 00:03:23.360 A:middle L:90%
the exponents. So 1/2 plus one is three house
49
00:03:23.370 --> 00:03:25.419 A:middle L:90%
, and then divide by this new exponents should just
50
00:03:25.419 --> 00:03:30.210 A:middle L:90%
result in 2/3. Um, and then we want
51
00:03:30.210 --> 00:03:34.210 A:middle L:90%
to evaluate this over our limits, which could be
52
00:03:35.340 --> 00:03:37.590 A:middle L:90%
pretty messy. When you're dealing with double into girls
53
00:03:38.240 --> 00:03:53.509 A:middle L:90%
. Let me write that. Okay. Um okay
54
00:03:53.520 --> 00:03:58.050 A:middle L:90%
, so now playing this in, you get 1/2
55
00:03:58.639 --> 00:04:08.990 A:middle L:90%
times 2/3 36 plus three y 3/2 and then minus
56
00:04:09.569 --> 00:04:11.590 A:middle L:90%
now, playing in the bottom of it, we
57
00:04:11.590 --> 00:04:15.850 A:middle L:90%
had 2/3 times nine plus three y all to the
58
00:04:15.860 --> 00:04:21.769 A:middle L:90%
three house power. Um, Now, what I'm
59
00:04:21.769 --> 00:04:26.629 A:middle L:90%
gonna do is just factor out this 2/3 so we
60
00:04:26.629 --> 00:04:29.500 A:middle L:90%
should get 1/3 on the outside, and then everything
61
00:04:29.500 --> 00:04:30.689 A:middle L:90%
else should remain the same. So we have 36
62
00:04:30.689 --> 00:04:34.680 A:middle L:90%
plus three y quantity to the three halves that minus
63
00:04:34.680 --> 00:04:39.269 A:middle L:90%
nine plus three y quantity to the three halves.
64
00:04:43.529 --> 00:04:47.259 A:middle L:90%
Um, now, this is just the inside internal
65
00:04:47.259 --> 00:04:49.769 A:middle L:90%
. Now we have to pluck this inside integral result
66
00:04:49.779 --> 00:04:54.579 A:middle L:90%
into the outside. Integral So we have been a
67
00:04:54.579 --> 00:04:58.079 A:middle L:90%
girl from 01 of 1/3 just going to take this
68
00:04:58.079 --> 00:05:03.639 A:middle L:90%
constant out right now of 36 plus three y 22
69
00:05:03.639 --> 00:05:08.699 A:middle L:90%
the three halfs minus nine plus three y quantity to
70
00:05:08.709 --> 00:05:15.740 A:middle L:90%
three halves de y. Now you may be thinking
71
00:05:15.740 --> 00:05:19.579 A:middle L:90%
that is little difficult to integrate this, but you'll
72
00:05:19.579 --> 00:05:27.379 A:middle L:90%
notice that basically, we have a linear function inside
73
00:05:27.389 --> 00:05:31.379 A:middle L:90%
of the power function or inside the power function.
74
00:05:31.589 --> 00:05:35.870 A:middle L:90%
So I'm going to use a little trick that comes
75
00:05:35.870 --> 00:05:39.600 A:middle L:90%
from you Substitution. But it's just a lot like
76
00:05:39.829 --> 00:05:42.800 A:middle L:90%
easier to do and just makes it a whole lot
77
00:05:42.800 --> 00:05:46.089 A:middle L:90%
faster. So we have 1/3 times the anti derivative
78
00:05:46.100 --> 00:05:49.459 A:middle L:90%
. So we're gonna treat this inside linear function as
79
00:05:49.459 --> 00:05:53.399 A:middle L:90%
if it were just like a Y, for example
80
00:05:54.319 --> 00:05:56.129 A:middle L:90%
. So, like why? To the three halves
81
00:05:56.220 --> 00:05:58.339 A:middle L:90%
. And if we were trying to anti Dreyfus,
82
00:05:59.639 --> 00:06:02.149 A:middle L:90%
we would get we would add once of the exponents
83
00:06:02.189 --> 00:06:03.819 A:middle L:90%
and then divide by that new exponents. So we
84
00:06:03.819 --> 00:06:06.730 A:middle L:90%
have 2/5 okay, and then plus C. But
85
00:06:06.740 --> 00:06:09.629 A:middle L:90%
we're doing it definitely. Girls said that doesn't really
86
00:06:09.629 --> 00:06:15.189 A:middle L:90%
matter. Um, so applying this concept to our
87
00:06:15.189 --> 00:06:18.769 A:middle L:90%
problem, we have 36 plus three y Also,
88
00:06:18.769 --> 00:06:23.509 A:middle L:90%
the five have his power on 2/5 because we're defined
89
00:06:23.509 --> 00:06:25.769 A:middle L:90%
by five house. So it just built location by
90
00:06:26.060 --> 00:06:29.300 A:middle L:90%
the reciprocal, which is 2/5. Then we wanted
91
00:06:29.300 --> 00:06:33.009 A:middle L:90%
to buy this entire thing by our coefficient linear coefficient
92
00:06:33.180 --> 00:06:40.810 A:middle L:90%
, which is just three. So the next one
93
00:06:40.810 --> 00:06:43.699 A:middle L:90%
would also use the same tactic because this is a
94
00:06:43.699 --> 00:06:46.879 A:middle L:90%
linear function here. So we have minus nine plus
95
00:06:46.879 --> 00:06:51.949 A:middle L:90%
three y toothy five house power no times to fifth
96
00:06:53.019 --> 00:06:56.459 A:middle L:90%
on the divide. All this by our linear coefficient
97
00:06:56.459 --> 00:07:00.509 A:middle L:90%
, which is just three we want Evaluate this from
98
00:07:00.509 --> 00:07:10.810 A:middle L:90%
0 to 1. So the reason, or basically
99
00:07:10.990 --> 00:07:14.069 A:middle L:90%
let me just verify that verify for you that,
100
00:07:14.209 --> 00:07:18.319 A:middle L:90%
um, this trick works. So if we look
101
00:07:18.319 --> 00:07:26.009 A:middle L:90%
at trying to derive this entire thing here, Well
102
00:07:26.019 --> 00:07:28.019 A:middle L:90%
, what we would first do is use the power
103
00:07:28.019 --> 00:07:30.829 A:middle L:90%
roll so we would bring this five halves down and
104
00:07:30.829 --> 00:07:33.069 A:middle L:90%
this would cancel with our two fists. Subtract that
105
00:07:33.079 --> 00:07:38.910 A:middle L:90%
new exponent by three halves or by 1/2 by Sorry
106
00:07:38.920 --> 00:07:42.449 A:middle L:90%
by one. And you get three house. And
107
00:07:42.449 --> 00:07:49.449 A:middle L:90%
then, um or actually, let me just write
108
00:07:49.449 --> 00:07:51.949 A:middle L:90%
this aside. So this is easier. Uh,
109
00:07:53.839 --> 00:07:57.360 A:middle L:90%
So if we were trying to take the derivative of
110
00:07:57.370 --> 00:08:11.139 A:middle L:90%
this. We were trying to derive this. Um
111
00:08:13.639 --> 00:08:15.689 A:middle L:90%
, I would first use the power rule. So
112
00:08:16.360 --> 00:08:22.029 A:middle L:90%
we have two times five or 2/5 times 36 plus
113
00:08:22.040 --> 00:08:26.910 A:middle L:90%
three. Y um, Now we take this five
114
00:08:26.910 --> 00:08:28.050 A:middle L:90%
house power, and it should cancel. If our
115
00:08:28.050 --> 00:08:31.740 A:middle L:90%
2/5 said those cancel, then we have three house
116
00:08:31.740 --> 00:08:35.980 A:middle L:90%
here. This is the by by three. And
117
00:08:35.980 --> 00:08:37.870 A:middle L:90%
then we have to use the train rule because we
118
00:08:37.870 --> 00:08:41.169 A:middle L:90%
have a function inside of another function. So because
119
00:08:41.169 --> 00:08:43.429 A:middle L:90%
the inside is a linear function, it's very easy
120
00:08:43.429 --> 00:08:46.149 A:middle L:90%
to take the dribble of that, and it should
121
00:08:46.149 --> 00:08:48.429 A:middle L:90%
just be three. So we multiply this by three
122
00:08:48.730 --> 00:08:50.279 A:middle L:90%
. It cancel it this and you would be left
123
00:08:50.279 --> 00:09:00.870 A:middle L:90%
with the original thing. Okay, Let me just
124
00:09:00.870 --> 00:09:05.470 A:middle L:90%
racist. Okay. So verifying that, um,
125
00:09:07.039 --> 00:09:09.259 A:middle L:90%
we can move on to after evaluating all this.
126
00:09:09.019 --> 00:09:16.549 A:middle L:90%
So we have 1/3 times plugging in one. We
127
00:09:16.549 --> 00:09:22.899 A:middle L:90%
should get two times 39 to the White House.
128
00:09:24.240 --> 00:09:26.440 A:middle L:90%
Bye bye. Five. And then divide this by
129
00:09:26.450 --> 00:09:33.980 A:middle L:90%
three than minus two times nine plus three should be
130
00:09:33.980 --> 00:09:37.970 A:middle L:90%
12. The five has five by five fold is
131
00:09:37.970 --> 00:09:41.029 A:middle L:90%
survived by three. Then we want to track all
132
00:09:41.029 --> 00:09:46.070 A:middle L:90%
this by what we plug in or the value when
133
00:09:46.070 --> 00:09:52.350 A:middle L:90%
we played in zero. So should be two times
134
00:09:52.360 --> 00:09:56.029 A:middle L:90%
36 plus three times zero is 36. It's a
135
00:09:56.039 --> 00:10:01.740 A:middle L:90%
five hives. Bye bye. Five to buy by
136
00:10:01.740 --> 00:10:07.899 A:middle L:90%
three. That minus two times. Um, nine
137
00:10:07.899 --> 00:10:13.370 A:middle L:90%
plus three times zero is 9 to£5 by by
138
00:10:13.370 --> 00:10:20.019 A:middle L:90%
five by three. All right, so once we
139
00:10:20.019 --> 00:10:24.950 A:middle L:90%
have this, um, well, we know that
140
00:10:24.960 --> 00:10:28.289 A:middle L:90%
, um let me just try to simplify this a
141
00:10:28.289 --> 00:10:31.799 A:middle L:90%
lot more before we move on. So, basically
142
00:10:31.809 --> 00:10:41.320 A:middle L:90%
, all these, um, components are being divided
143
00:10:41.320 --> 00:10:46.940 A:middle L:90%
by three. So I'm just gonna pull out the
144
00:10:46.940 --> 00:10:50.750 A:middle L:90%
1/3 first. So we have 1/9 on the outside
145
00:10:52.769 --> 00:10:56.950 A:middle L:90%
, then we have 2/5 left on each component.
146
00:10:58.440 --> 00:11:07.379 A:middle L:90%
So let me write that minus two times 12 5
147
00:11:07.389 --> 00:11:13.320 A:middle L:90%
halves over five. And then since we're subtracting this
148
00:11:13.320 --> 00:11:16.309 A:middle L:90%
country, be negative, and this should be positive
149
00:11:16.320 --> 00:11:22.490 A:middle L:90%
. So minus two fists, 36 5 hives and
150
00:11:22.490 --> 00:11:28.389 A:middle L:90%
then plus two times nine to If I have over
151
00:11:28.399 --> 00:11:33.470 A:middle L:90%
five now, I'm gonna factor out the two fists
152
00:11:33.480 --> 00:11:37.720 A:middle L:90%
, so we should get to 45th on the outside
153
00:11:37.830 --> 00:11:41.350 A:middle L:90%
because two times one is two and 19 5 45
154
00:11:41.340 --> 00:11:43.940 A:middle L:90%
So now the in shy inside should look a lot
155
00:11:43.940 --> 00:11:48.409 A:middle L:90%
nicer. So 39 um, to the five halves
156
00:11:48.419 --> 00:11:56.470 A:middle L:90%
, minus 12 to 5 halves and then minus 36
157
00:11:56.480 --> 00:12:07.669 A:middle L:90%
5 halves plus 9 to 5 halves. Um,
158
00:12:09.259 --> 00:12:16.769 A:middle L:90%
make sure. Okay, then, once we're at
159
00:12:16.769 --> 00:12:20.799 A:middle L:90%
this part, um, we're gonna keep the 1st
160
00:12:20.799 --> 00:12:28.509 A:middle L:90%
2 components in there in simplified form. Because if
161
00:12:28.509 --> 00:12:31.350 A:middle L:90%
you tried some file result a dustman, um,
162
00:12:31.840 --> 00:12:35.409 A:middle L:90%
this part So applying the to to both of these
163
00:12:35.409 --> 00:12:39.669 A:middle L:90%
numbers on the inside, we're basically just square rooting
164
00:12:39.669 --> 00:12:41.149 A:middle L:90%
them. So if I were to rewrite, thes
165
00:12:41.840 --> 00:12:48.460 A:middle L:90%
would be basically the square root of 36 and then
166
00:12:48.460 --> 00:12:50.940 A:middle L:90%
all of this fifth power, and then this would
167
00:12:50.940 --> 00:12:54.710 A:middle L:90%
be square of nine all to the fifth Power.
168
00:12:56.639 --> 00:13:00.379 A:middle L:90%
And we know the square root of both of those
169
00:13:00.379 --> 00:13:03.210 A:middle L:90%
numbers so we can simplify that a little bit.
170
00:13:07.549 --> 00:13:09.559 A:middle L:90%
The square 36 that's 66 the fifth, and then
171
00:13:09.559 --> 00:13:16.379 A:middle L:90%
we have plus three two on this monstrous number is
172
00:13:16.389 --> 00:13:16.610 A:middle L:90%
our final answer.