WEBVTT
1
00:00:05.740 --> 00:00:10.839 A:middle L:90%
we want to find a volume that is generated by
2
00:00:10.839 --> 00:00:17.850 A:middle L:90%
rotating the region bounded by the curve. Why equals
3
00:00:17.859 --> 00:00:22.769 A:middle L:90%
uh E to the negative X squared. And uh
4
00:00:22.780 --> 00:00:27.440 A:middle L:90%
the lines Y equals zero, X equals zero And
5
00:00:27.449 --> 00:00:32.020 A:middle L:90%
Ux x equals one. Uh So the region that
6
00:00:32.020 --> 00:00:34.929 A:middle L:90%
is bounded by this curb. And these lines,
7
00:00:34.939 --> 00:00:38.710 A:middle L:90%
that region is going to be rotated around the y
8
00:00:38.710 --> 00:00:41.820 A:middle L:90%
axis. And we want to use the method of
9
00:00:41.820 --> 00:00:45.719 A:middle L:90%
cylindrical shells to find the volume generated to get a
10
00:00:45.719 --> 00:00:48.820 A:middle L:90%
better idea idea of what this region looks like.
11
00:00:48.820 --> 00:00:51.950 A:middle L:90%
I have a uh this curve and these three lines
12
00:00:51.950 --> 00:00:55.710 A:middle L:90%
graft on the dez most graphing calculator. All right
13
00:00:55.710 --> 00:00:58.869 A:middle L:90%
, so here's the portion of e. Actually it's
14
00:00:58.869 --> 00:01:06.549 A:middle L:90%
craft incorrectly. Let me fix this. We raised
15
00:01:06.939 --> 00:01:11.150 A:middle L:90%
to the negative X squared. There we go.
16
00:01:14.439 --> 00:01:18.709 A:middle L:90%
Okay. Uh So here's the portion of the curve
17
00:01:18.709 --> 00:01:23.370 A:middle L:90%
Y equals E to the negative X square between ex
18
00:01:23.379 --> 00:01:26.269 A:middle L:90%
uh equals zero and X equals one. And of
19
00:01:26.269 --> 00:01:29.349 A:middle L:90%
course this is the line Y equals zero. So
20
00:01:29.349 --> 00:01:30.980 A:middle L:90%
we're looking at this little region in here, let's
21
00:01:30.980 --> 00:01:33.890 A:middle L:90%
blow it up a little bit. We're looking at
22
00:01:33.890 --> 00:01:38.730 A:middle L:90%
this region in here uh is going to be rotated
23
00:01:38.739 --> 00:01:41.769 A:middle L:90%
around this y axis. So we're going to take
24
00:01:41.769 --> 00:01:45.219 A:middle L:90%
this region and spin it around the y axis.
25
00:01:45.230 --> 00:01:48.079 A:middle L:90%
And we want to use the method of cylindrical shells
26
00:01:48.079 --> 00:01:53.430 A:middle L:90%
to find that volume. Well volume. Using a
27
00:01:53.430 --> 00:01:59.790 A:middle L:90%
cylindrical shells is going to equal to times pi time
28
00:01:59.790 --> 00:02:04.280 A:middle L:90%
she entered role. Looking at the uh region.
29
00:02:04.280 --> 00:02:07.750 A:middle L:90%
Again, we're going to integrate from X zero up
30
00:02:07.750 --> 00:02:15.449 A:middle L:90%
to x is one uh X times our function.
31
00:02:15.449 --> 00:02:17.849 A:middle L:90%
Why? Which is E. To the negative X
32
00:02:17.849 --> 00:02:23.770 A:middle L:90%
squared D. X. So once we calculate this
33
00:02:23.770 --> 00:02:28.090 A:middle L:90%
definite integral uh That will be the volume when this
34
00:02:28.090 --> 00:02:32.199 A:middle L:90%
region is rotated around the y axis. And this
35
00:02:32.199 --> 00:02:37.969 A:middle L:90%
is using the method of cylindrical shells. Well to
36
00:02:37.969 --> 00:02:45.560 A:middle L:90%
evaluate this integral. Uh First um let's go back
37
00:02:45.560 --> 00:02:52.069 A:middle L:90%
to let's just go off to the side here.
38
00:02:52.080 --> 00:02:55.250 A:middle L:90%
What is the integral of X. He to the
39
00:02:55.250 --> 00:03:00.710 A:middle L:90%
negative X squared D. X. If we use
40
00:03:00.719 --> 00:03:07.680 A:middle L:90%
U substitution let U equal negative X squared. Then
41
00:03:07.680 --> 00:03:09.680 A:middle L:90%
D. U. D. X Would be-2
42
00:03:09.680 --> 00:03:15.659 A:middle L:90%
X. Uh so d'You would be-2 X.
43
00:03:16.840 --> 00:03:21.360 A:middle L:90%
D. X. Let's get a space here.
44
00:03:23.539 --> 00:03:28.439 A:middle L:90%
So I wanna rewrite this integral in terms of use
45
00:03:28.449 --> 00:03:43.270 A:middle L:90%
now this will be each of you. So in
46
00:03:43.270 --> 00:03:47.819 A:middle L:90%
a girl. Alright each negative X squared. Um
47
00:03:47.830 --> 00:03:51.849 A:middle L:90%
Each a negative X squared will be E. To
48
00:03:51.849 --> 00:03:57.159 A:middle L:90%
do you. No X. D. X.
49
00:03:57.840 --> 00:04:03.330 A:middle L:90%
Yeah is equal to-1/2. You. Uh So
50
00:04:03.330 --> 00:04:05.159 A:middle L:90%
if we want to put A D. U.
51
00:04:05.159 --> 00:04:08.960 A:middle L:90%
In here we need negative two X. Dx.
52
00:04:09.639 --> 00:04:12.419 A:middle L:90%
So if we want a negative two X. D
53
00:04:12.419 --> 00:04:15.439 A:middle L:90%
. X. So that we can write negative two
54
00:04:15.439 --> 00:04:18.870 A:middle L:90%
X. D. X. As D. U
55
00:04:18.879 --> 00:04:21.459 A:middle L:90%
. That means I want to put a negative too
56
00:04:25.339 --> 00:04:30.490 A:middle L:90%
negative two. Let me write this negative two has
57
00:04:30.490 --> 00:04:33.620 A:middle L:90%
two times this. But if I'm time seeing this
58
00:04:33.620 --> 00:04:36.329 A:middle L:90%
by negative two we have to balance it by multiplying
59
00:04:36.339 --> 00:04:40.699 A:middle L:90%
out here by negative one half. Negative one half
60
00:04:40.699 --> 00:04:43.129 A:middle L:90%
times negative two is positive one. So I'm not
61
00:04:43.129 --> 00:04:46.439 A:middle L:90%
changing anything by multiplying by one. But the E
62
00:04:46.449 --> 00:04:49.319 A:middle L:90%
. To the negative X squared is E. To
63
00:04:49.319 --> 00:04:51.779 A:middle L:90%
the U. Uh Since negative X squared is my
64
00:04:51.779 --> 00:04:54.500 A:middle L:90%
U. E. To the negative X squared is
65
00:04:54.509 --> 00:04:57.000 A:middle L:90%
E. To the U. The negative two X
66
00:04:57.000 --> 00:05:00.089 A:middle L:90%
times dx That's my D. U. And then
67
00:05:00.089 --> 00:05:02.459 A:middle L:90%
of course I have the ties by negative 1/2.
68
00:05:08.040 --> 00:05:15.100 A:middle L:90%
So this integral is really uh the integral of uh
69
00:05:15.110 --> 00:05:18.379 A:middle L:90%
X. E. To the negative X squared dx
70
00:05:18.379 --> 00:05:20.930 A:middle L:90%
. This is the hardest part of the problem.
71
00:05:20.930 --> 00:05:27.620 A:middle L:90%
Not that it's hard but when I find this integral
72
00:05:27.620 --> 00:05:30.629 A:middle L:90%
I'm basically finding the integral of X. Each negative
73
00:05:30.629 --> 00:05:32.819 A:middle L:90%
X squared dx. So I'll know what the anti
74
00:05:32.819 --> 00:05:39.529 A:middle L:90%
derivative of X eaten negative X squared is. Once
75
00:05:39.529 --> 00:05:42.060 A:middle L:90%
I know the anti derivative, then I just had
76
00:05:42.060 --> 00:05:45.459 A:middle L:90%
to evaluate it between the limits of X X one
77
00:05:45.470 --> 00:05:47.600 A:middle L:90%
and x zero. So once I calculate this I'm
78
00:05:47.600 --> 00:05:51.060 A:middle L:90%
going to get an expression uh in terms of you
79
00:05:51.439 --> 00:05:55.680 A:middle L:90%
and then I'll substitute back in what U. Equals
80
00:05:55.680 --> 00:05:59.300 A:middle L:90%
. So then I'll have uh an expression in terms
81
00:05:59.300 --> 00:06:01.240 A:middle L:90%
of X. Which will basically be the anti derivative
82
00:06:01.250 --> 00:06:05.259 A:middle L:90%
of X. Each negative x squared dx. So
83
00:06:06.439 --> 00:06:09.899 A:middle L:90%
Uh integral v. To you. That's an easy
84
00:06:09.899 --> 00:06:13.430 A:middle L:90%
one. The enemy derivative of EUU. Is each
85
00:06:13.430 --> 00:06:15.649 A:middle L:90%
to you. So this is equal to negative 1/2
86
00:06:17.540 --> 00:06:24.310 A:middle L:90%
E. To do you Which is really equal to
87
00:06:24.370 --> 00:06:30.850 A:middle L:90%
negative 1/2. Uh Since you is negative X squared
88
00:06:30.850 --> 00:06:32.040 A:middle L:90%
negative one half. Each of the eu is negative
89
00:06:32.040 --> 00:06:36.250 A:middle L:90%
one half E. To the negative X squared.
90
00:06:41.540 --> 00:06:43.930 A:middle L:90%
So not our final answer. But this is the
91
00:06:43.930 --> 00:06:47.259 A:middle L:90%
anti derivative of X. Each negative X squared.
92
00:06:47.839 --> 00:06:51.209 A:middle L:90%
If you took the derivative of this, if you
93
00:06:51.209 --> 00:06:53.720 A:middle L:90%
want to check your work, if you take the
94
00:06:53.720 --> 00:06:57.589 A:middle L:90%
derivative of negative 1/2 times E. To the negative
95
00:06:57.589 --> 00:07:00.550 A:middle L:90%
X squared, you will get X E to the
96
00:07:00.550 --> 00:07:02.189 A:middle L:90%
negative X square. So negative one half E.
97
00:07:02.189 --> 00:07:05.639 A:middle L:90%
To the negative X square is the anti derivative of
98
00:07:05.639 --> 00:07:12.089 A:middle L:90%
this. So running out of space, I'm going
99
00:07:12.089 --> 00:07:16.170 A:middle L:90%
to rewrite this uh volume formula. Uh Using cylindrical
100
00:07:16.170 --> 00:07:18.470 A:middle L:90%
shells rotating around the y axis. I'm going to
101
00:07:18.470 --> 00:07:23.959 A:middle L:90%
rewrite this formula down here and we're actually almost done
102
00:07:30.240 --> 00:07:33.829 A:middle L:90%
. Okay. The volume in that region is rotated
103
00:07:33.839 --> 00:07:38.459 A:middle L:90%
uh around the y axis. Using the method of
104
00:07:38.459 --> 00:07:43.050 A:middle L:90%
cylindrical shells is equal to two pi Times the integral
105
00:07:43.050 --> 00:07:49.560 A:middle L:90%
from 0 to 1. Uh E it was X
106
00:07:49.569 --> 00:07:54.649 A:middle L:90%
. E. To the negative X squared D X
107
00:07:55.339 --> 00:07:57.970 A:middle L:90%
. So this will be the volume of the region
108
00:07:57.970 --> 00:08:01.240 A:middle L:90%
. Once it's rotated around the y axis. Now
109
00:08:01.250 --> 00:08:03.319 A:middle L:90%
, the anti derivative of X. E. To
110
00:08:03.319 --> 00:08:07.519 A:middle L:90%
the negative x square is uh this expression right here
111
00:08:07.529 --> 00:08:11.240 A:middle L:90%
in the red box. So this volume is simply
112
00:08:11.240 --> 00:08:18.610 A:middle L:90%
going to be two pi times The anti derivative.
113
00:08:18.620 --> 00:08:24.250 A:middle L:90%
The negative 1/2 E. To the negative X squared
114
00:08:26.040 --> 00:08:33.840 A:middle L:90%
uh evaluated between X is one And X0. So
115
00:08:33.840 --> 00:08:37.549 A:middle L:90%
basically we have to uh calculate the value of this
116
00:08:37.549 --> 00:08:39.559 A:middle L:90%
definite integral. So we take it the anti derivative
117
00:08:41.340 --> 00:08:43.980 A:middle L:90%
and we evaluated at one and then we subtract the
118
00:08:43.980 --> 00:08:48.350 A:middle L:90%
same anti derivative evaluated at zero. Now just to
119
00:08:48.350 --> 00:08:50.509 A:middle L:90%
make life a little easier, we could do the
120
00:08:50.509 --> 00:08:54.269 A:middle L:90%
two times in one half and we can do that
121
00:08:54.279 --> 00:08:56.159 A:middle L:90%
and you'll still have this negative. So let's let's
122
00:08:56.159 --> 00:08:58.720 A:middle L:90%
take this one negative one half. That's times and
123
00:08:58.720 --> 00:09:01.460 A:middle L:90%
all this. And let's move it out here just
124
00:09:01.460 --> 00:09:03.840 A:middle L:90%
to make things a little bit easier. So two
125
00:09:03.840 --> 00:09:07.759 A:middle L:90%
pi times negative one half will be negative pi.
126
00:09:07.539 --> 00:09:11.830 A:middle L:90%
So we really just have to take negative by in
127
00:09:11.830 --> 00:09:16.980 A:middle L:90%
times of by E to the negative X squared Evaluated
128
00:09:16.980 --> 00:09:22.379 A:middle L:90%
between one and 0. Well that's going to equal
129
00:09:22.379 --> 00:09:26.690 A:middle L:90%
negative pi times while each of the negative X squared
130
00:09:26.690 --> 00:09:28.629 A:middle L:90%
when X is one is E. To the negative
131
00:09:28.629 --> 00:09:33.919 A:middle L:90%
one. Then we have to subtract e two negative
132
00:09:33.919 --> 00:09:35.940 A:middle L:90%
x squared evaluated at zero, which would be E
133
00:09:35.940 --> 00:09:39.269 A:middle L:90%
. To the negative zero square T. 20 which
134
00:09:39.269 --> 00:09:50.080 A:middle L:90%
is one. So uh multiplying negative pi by each
135
00:09:50.080 --> 00:09:54.259 A:middle L:90%
of these uh negative pi times negative one is pie
136
00:09:56.139 --> 00:10:00.889 A:middle L:90%
negative pi times E. To the negative one is
137
00:10:00.889 --> 00:10:03.549 A:middle L:90%
a negative pi. Each a negative one but each
138
00:10:03.549 --> 00:10:05.789 A:middle L:90%
a negative one is 1/8. So we can write
139
00:10:05.789 --> 00:10:09.490 A:middle L:90%
this as minus pi Over each of the one or
140
00:10:09.500 --> 00:10:15.029 A:middle L:90%
8. So there is our volume when we rotated
141
00:10:15.029 --> 00:10:18.309 A:middle L:90%
that region around the y axis. Using the method
142
00:10:18.320 --> 00:10:20.860 A:middle L:90%
of cylindrical shells