WEBVTT
1
00:00:01.240 --> 00:00:02.879 A:middle L:90%
We want to do a quick sketch of these two
2
00:00:02.879 --> 00:00:05.450 A:middle L:90%
functions and then find the area of the enclosed region
3
00:00:06.040 --> 00:00:07.750 A:middle L:90%
. So let's take a look at the 1st 1
4
00:00:08.179 --> 00:00:12.070 A:middle L:90%
Let's draw it in green, so it's long necks
5
00:00:12.070 --> 00:00:14.730 A:middle L:90%
over X. We know that lawn expert grows very
6
00:00:14.730 --> 00:00:18.250 A:middle L:90%
slowly so that as X goes to infinity, um
7
00:00:18.390 --> 00:00:20.420 A:middle L:90%
, it's, uh, the function will go to
8
00:00:20.420 --> 00:00:23.250 A:middle L:90%
zero s o the end behavior here will boat a
9
00:00:23.250 --> 00:00:27.710 A:middle L:90%
zero. Um, but for smaller X did,
10
00:00:27.780 --> 00:00:32.479 A:middle L:90%
um, looks like lawn of X. So that's
11
00:00:32.479 --> 00:00:35.310 A:middle L:90%
what lan of X over X looks like. Now
12
00:00:35.310 --> 00:00:39.960 A:middle L:90%
let's draw the 2nd 1 in blue a lot of
13
00:00:39.960 --> 00:00:42.649 A:middle L:90%
X squared. So here we're alone. X is
14
00:00:42.649 --> 00:00:44.979 A:middle L:90%
going to be positive. So we're actually going from
15
00:00:44.979 --> 00:00:53.679 A:middle L:90%
appear and it goes like that. So we see
16
00:00:53.679 --> 00:00:56.329 A:middle L:90%
that we have this enclosed region and we want to
17
00:00:56.329 --> 00:00:59.280 A:middle L:90%
figure out what our limits of integration. So it's
18
00:00:59.280 --> 00:01:06.209 A:middle L:90%
a sulfur that we just equate the two equations lawn
19
00:01:06.219 --> 00:01:15.310 A:middle L:90%
X over X equals lan X squared over X.
20
00:01:15.040 --> 00:01:19.379 A:middle L:90%
So we see that there's two possibilities. Ah,
21
00:01:19.489 --> 00:01:22.790 A:middle L:90%
In one case, we have lawn X equals zero
22
00:01:23.400 --> 00:01:26.469 A:middle L:90%
, which gives us X is equal to one.
23
00:01:26.739 --> 00:01:30.099 A:middle L:90%
So that's one solution in the other case, Lawn
24
00:01:30.099 --> 00:01:32.980 A:middle L:90%
of X is not equal to zero so that we
25
00:01:32.980 --> 00:01:34.549 A:middle L:90%
can divide by both a bylaw next on both sides
26
00:01:36.180 --> 00:01:38.569 A:middle L:90%
. So in this case, we can cancel one
27
00:01:38.689 --> 00:01:42.650 A:middle L:90%
lawn X. So this case will be in red
28
00:01:42.040 --> 00:01:45.150 A:middle L:90%
. This allows us to cancel one lung X.
29
00:01:46.930 --> 00:01:49.049 A:middle L:90%
So then what we have is one over X equals
30
00:01:49.590 --> 00:01:56.030 A:middle L:90%
lawn X over X. We can cancel these exes
31
00:01:56.030 --> 00:02:00.890 A:middle L:90%
so we just have long X equals one or raising
32
00:02:00.890 --> 00:02:02.299 A:middle L:90%
both sides by e So we have e to the
33
00:02:02.299 --> 00:02:07.379 A:middle L:90%
one is equal to X. So that's our second
34
00:02:07.379 --> 00:02:12.030 A:middle L:90%
point. So we want we're going to be integrating
35
00:02:12.030 --> 00:02:16.150 A:middle L:90%
from this left most point here, X equals one
36
00:02:16.639 --> 00:02:20.430 A:middle L:90%
to this point. Over here, X equals E
37
00:02:20.530 --> 00:02:24.370 A:middle L:90%
, which is about 2.7. So let's carry the
38
00:02:24.370 --> 00:02:30.219 A:middle L:90%
integral to figure out the area area is equal to
39
00:02:30.300 --> 00:02:35.039 A:middle L:90%
integral. We're going from one to e of art
40
00:02:35.050 --> 00:02:37.789 A:middle L:90%
top function. Here. The top function was the
41
00:02:37.789 --> 00:02:38.750 A:middle L:90%
green one. So that's the lawn X over Rex
42
00:02:39.240 --> 00:02:42.449 A:middle L:90%
. And the bottom function is lan X squared over
43
00:02:42.449 --> 00:02:46.930 A:middle L:90%
X. So we're going to subtract one x over
44
00:02:46.009 --> 00:02:55.360 A:middle L:90%
X minus lawn X squared over X DX to sulfur
45
00:02:55.360 --> 00:02:58.889 A:middle L:90%
. This, um we're going to do a U
46
00:02:58.889 --> 00:03:01.650 A:middle L:90%
substitution. Let's actually write these over the same.
47
00:03:02.639 --> 00:03:07.750 A:middle L:90%
Uh, okay, this is her a minus here
48
00:03:13.590 --> 00:03:15.750 A:middle L:90%
, right? Because we can just combine the fractions
49
00:03:15.439 --> 00:03:19.360 A:middle L:90%
. So there we go. Now we can do
50
00:03:19.360 --> 00:03:25.030 A:middle L:90%
a U Substitution of U equals Lawn X, and
51
00:03:25.030 --> 00:03:30.189 A:middle L:90%
this gives us d'you equals one over x d x
52
00:03:30.259 --> 00:03:34.689 A:middle L:90%
. It's already we see here this d x over
53
00:03:34.689 --> 00:03:38.110 A:middle L:90%
X, that's are do you and we can just
54
00:03:38.120 --> 00:03:43.960 A:middle L:90%
plug in our you Fairlawn X. So of course
55
00:03:43.960 --> 00:03:45.849 A:middle L:90%
we're going to need our new limits of integration.
56
00:03:46.340 --> 00:03:47.669 A:middle L:90%
So we see that when x are so looking over
57
00:03:47.669 --> 00:03:51.599 A:middle L:90%
here when X is one, we know that you
58
00:03:51.599 --> 00:03:54.400 A:middle L:90%
was zero. And when ex's e we have u
59
00:03:54.400 --> 00:03:57.310 A:middle L:90%
equals lot of e, which is just one.
60
00:03:58.490 --> 00:04:00.810 A:middle L:90%
Okay, so now we have lawn X, which
61
00:04:00.810 --> 00:04:04.129 A:middle L:90%
is this you minus lan X squared. So you
62
00:04:04.129 --> 00:04:08.090 A:middle L:90%
minus you squared and in D X over exes are
63
00:04:08.090 --> 00:04:10.770 A:middle L:90%
do you so we can sell for this? But
64
00:04:10.870 --> 00:04:15.710 A:middle L:90%
, uh, quite simply, the anti derivative of
65
00:04:15.710 --> 00:04:18.920 A:middle L:90%
you is half you squared and two driven off you
66
00:04:18.920 --> 00:04:23.069 A:middle L:90%
squared is one over three, you cubed. We're
67
00:04:23.069 --> 00:04:25.759 A:middle L:90%
going from 0 to 1. So we just put
68
00:04:25.759 --> 00:04:29.370 A:middle L:90%
in our numbers. We have half times one minus
69
00:04:29.370 --> 00:04:34.209 A:middle L:90%
1/3 times one minus zero minus zero, which is
70
00:04:34.209 --> 00:04:40.519 A:middle L:90%
equal to half minus 1/3 which is 1/6 so the
71
00:04:40.519 --> 00:04:42.850 A:middle L:90%
area of the enclosed region is one over six.