WEBVTT
1
00:00:01.040 --> 00:00:05.110 A:middle L:90%
So if this question were passed to determine the lining
2
00:00:05.110 --> 00:00:09.660 A:middle L:90%
to grow Uh, X Y yes. And we're
3
00:00:09.660 --> 00:00:12.400 A:middle L:90%
given that the curve. We have a pyramid to
4
00:00:12.400 --> 00:00:17.129 A:middle L:90%
repair amateur eyes curve where X is equal squared and
5
00:00:17.129 --> 00:00:20.219 A:middle L:90%
wise. He would take two to t and we're
6
00:00:20.219 --> 00:00:25.890 A:middle L:90%
told that goes from zero 21 All right, so
7
00:00:25.890 --> 00:00:27.890 A:middle L:90%
the first thing we're gonna do is we're going to
8
00:00:27.899 --> 00:00:32.090 A:middle L:90%
determine or clank for DS. Then we know that
9
00:00:32.270 --> 00:00:37.600 A:middle L:90%
the S is square Heard of d y by d
10
00:00:37.609 --> 00:00:42.189 A:middle L:90%
Squared plus DX by DT square TT So if we
11
00:00:42.189 --> 00:00:45.170 A:middle L:90%
take why and we finest derivative with respect it about
12
00:00:45.179 --> 00:00:48.369 A:middle L:90%
activity, so lies just to t so that the
13
00:00:48.369 --> 00:00:51.579 A:middle L:90%
derivative of, uh, the derivative of that with
14
00:00:51.590 --> 00:00:55.770 A:middle L:90%
respect to, uh, t so to t the
15
00:00:55.770 --> 00:00:58.750 A:middle L:90%
derivative of to tea with respect a tease just to
16
00:00:59.840 --> 00:01:03.329 A:middle L:90%
And then also we take a riveting vax with respect
17
00:01:03.340 --> 00:01:06.480 A:middle L:90%
. Cities of excess just t square. The derivative
18
00:01:06.480 --> 00:01:10.519 A:middle L:90%
of that with respected tea is simply too All right
19
00:01:10.519 --> 00:01:15.500 A:middle L:90%
. So finally you get square root of two squared
20
00:01:15.590 --> 00:01:22.959 A:middle L:90%
plus two. So this just four plus 40 squared
21
00:01:22.969 --> 00:01:26.209 A:middle L:90%
. We can pull out of forest common factor the
22
00:01:26.209 --> 00:01:29.879 A:middle L:90%
square root of forest to And then what's left inside
23
00:01:29.879 --> 00:01:32.450 A:middle L:90%
this square and square root is just X squared.
24
00:01:33.739 --> 00:01:37.890 A:middle L:90%
So we determined our ts, our ideas we have
25
00:01:37.900 --> 00:01:40.189 A:middle L:90%
we know that access t squared. We know that
26
00:01:40.200 --> 00:01:42.950 A:middle L:90%
. Why is two teeth so simply what we're gonna
27
00:01:42.950 --> 00:01:45.870 A:middle L:90%
do is we're gonna plunk now everything back into this
28
00:01:45.870 --> 00:01:53.030 A:middle L:90%
and try to determine. Okay, so now when
29
00:01:53.030 --> 00:01:56.030 A:middle L:90%
we plug everything back, we get this big mess
30
00:01:56.030 --> 00:01:57.319 A:middle L:90%
right here, but we're gonna try that still,
31
00:01:57.840 --> 00:02:00.030 A:middle L:90%
So we pull out the tour. So two times
32
00:02:00.030 --> 00:02:01.959 A:middle L:90%
two is four so we can pull that out as
33
00:02:01.959 --> 00:02:06.939 A:middle L:90%
a constant And what we're left with tears. That's
34
00:02:06.939 --> 00:02:10.199 A:middle L:90%
right. So we haven't too sweaty square, plus
35
00:02:10.199 --> 00:02:13.180 A:middle L:90%
one under the square root, and we have a
36
00:02:13.240 --> 00:02:15.310 A:middle L:90%
cubed on the outside. Well, let's try to
37
00:02:15.650 --> 00:02:20.509 A:middle L:90%
solve this integral using substitution. So we're gonna sit
38
00:02:20.520 --> 00:02:23.449 A:middle L:90%
you equal to the square, plus work, and
39
00:02:23.460 --> 00:02:25.419 A:middle L:90%
then if we take the derivative of there we get
40
00:02:25.509 --> 00:02:29.430 A:middle L:90%
to you is to teach. And then if we
41
00:02:29.430 --> 00:02:31.110 A:middle L:90%
divide by two on both sides, we get that
42
00:02:31.120 --> 00:02:35.250 A:middle L:90%
TV team is equal to you, divided by two
43
00:02:36.039 --> 00:02:38.319 A:middle L:90%
. Also, we're gonna do something that might be
44
00:02:38.319 --> 00:02:40.710 A:middle L:90%
a bit unusual, but, you know, we
45
00:02:40.710 --> 00:02:44.840 A:middle L:90%
can also since we said u equals C squared plus
46
00:02:44.840 --> 00:02:46.590 A:middle L:90%
one. What we can do is we noticed that
47
00:02:46.599 --> 00:02:49.990 A:middle L:90%
he squared. Is he good to you? Minus
48
00:02:49.990 --> 00:02:52.479 A:middle L:90%
one. So no, this is create because that
49
00:02:52.479 --> 00:02:55.780 A:middle L:90%
we can write everything in the integral here in terms
50
00:02:55.780 --> 00:02:59.900 A:middle L:90%
of you. So also, we were gonna change
51
00:02:59.909 --> 00:03:01.909 A:middle L:90%
the limits so the lower limit was zero, but
52
00:03:01.909 --> 00:03:06.289 A:middle L:90%
that we know that there's a mystical tissue we can
53
00:03:06.289 --> 00:03:09.689 A:middle L:90%
pluck that into t squared+10 square plus one is
54
00:03:09.689 --> 00:03:12.939 A:middle L:90%
just one. So that's our new limit. Because
55
00:03:12.939 --> 00:03:15.659 A:middle L:90%
we want everything in terms of you and again,
56
00:03:15.659 --> 00:03:16.819 A:middle L:90%
we have our upper limit off. The integral is
57
00:03:16.830 --> 00:03:20.289 A:middle L:90%
teasing cool toe one. So if we plug that
58
00:03:20.289 --> 00:03:22.949 A:middle L:90%
into use equal to the square Plus when we get
59
00:03:22.949 --> 00:03:25.849 A:middle L:90%
one square, plus one more to So now we
60
00:03:25.849 --> 00:03:30.099 A:middle L:90%
completely then we have everything in terms of you.
61
00:03:30.199 --> 00:03:32.789 A:middle L:90%
So everything we have in terms of you and R
62
00:03:32.789 --> 00:03:36.099 A:middle L:90%
d t is just Do you divided by two so
63
00:03:36.099 --> 00:03:38.139 A:middle L:90%
we could pull that tutor the outside So four divided
64
00:03:38.150 --> 00:03:42.509 A:middle L:90%
by two is just all right. So now we
65
00:03:42.509 --> 00:03:46.759 A:middle L:90%
have to take the integral, uh U to the
66
00:03:46.759 --> 00:03:51.460 A:middle L:90%
power of 3/2 minus you did well. The integral
67
00:03:51.460 --> 00:03:54.020 A:middle L:90%
beauty, the power of three divided by two,
68
00:03:54.020 --> 00:03:59.599 A:middle L:90%
is just due to the power 5/2 times Tool and
69
00:03:59.599 --> 00:04:01.379 A:middle L:90%
the integral of U to the power half is disputed
70
00:04:01.389 --> 00:04:04.860 A:middle L:90%
. The power 3/2 times two of us. And
71
00:04:04.860 --> 00:04:08.110 A:middle L:90%
the limit is from 2 to 1 of these.
72
00:04:10.240 --> 00:04:12.939 A:middle L:90%
So now we just plug everything back in. We
73
00:04:12.939 --> 00:04:15.050 A:middle L:90%
don't forget we have a two on the outside.
74
00:04:15.639 --> 00:04:16.829 A:middle L:90%
But then if we work out the out abruptly put
75
00:04:16.829 --> 00:04:19.149 A:middle L:90%
this into our calculator, we're gonna get that.
76
00:04:19.149 --> 00:04:26.660 A:middle L:90%
Our final answer is 1.2876 So that's the value of
77
00:04:26.670 --> 00:04:27.220 A:middle L:90%
the line.