WEBVTT
1
00:00:01.940 --> 00:00:06.559 A:middle L:90%
So here we have an integral from 0 to 1
2
00:00:07.040 --> 00:00:11.259 A:middle L:90%
of X squared over one plus X to the fourth
3
00:00:11.929 --> 00:00:15.140 A:middle L:90%
. We're and equals 10. Um, and our
4
00:00:15.150 --> 00:00:18.160 A:middle L:90%
interval with Delta X is going to be equal to
5
00:00:18.170 --> 00:00:21.260 A:middle L:90%
B minus a over end. So one minus 0/10
6
00:00:21.640 --> 00:00:26.289 A:middle L:90%
, which is your 0.1. So using the trapezoidal
7
00:00:26.289 --> 00:00:29.059 A:middle L:90%
rule, what we'll have is Delta X over two
8
00:00:29.839 --> 00:00:32.090 A:middle L:90%
. Um, it's gonna look like Delta X over
9
00:00:32.090 --> 00:00:39.229 A:middle L:90%
two times F of zero plus two times F of
10
00:00:39.229 --> 00:00:46.810 A:middle L:90%
0.1 plus two times F of 0.2 plus two times
11
00:00:46.909 --> 00:00:56.359 A:middle L:90%
F of 0.3 plus two times F of 0.4 plus
12
00:00:56.740 --> 00:01:03.109 A:middle L:90%
two times F of 0.5 plus two times f of
13
00:01:03.189 --> 00:01:07.799 A:middle L:90%
six You get the picture plus seven eight, all
14
00:01:07.799 --> 00:01:10.849 A:middle L:90%
right, and then lastly, we'll have to f
15
00:01:10.849 --> 00:01:19.540 A:middle L:90%
of 0.9 plus f of 0.1. When we do
16
00:01:19.540 --> 00:01:23.359 A:middle L:90%
all that, what we end up getting Delta X
17
00:01:23.359 --> 00:01:26.879 A:middle L:90%
is obviously 0.1. So I've point 1/2. What
18
00:01:26.879 --> 00:01:34.959 A:middle L:90%
we end up getting as a result is approximately 0.243747
19
00:01:34.340 --> 00:01:44.680 A:middle L:90%
for our t 10, then for part B we
20
00:01:44.689 --> 00:01:49.530 A:middle L:90%
want to do the same method. Only this time
21
00:01:49.530 --> 00:01:51.840 A:middle L:90%
it's going to be interval in points. So what
22
00:01:51.840 --> 00:01:56.909 A:middle L:90%
that's gonna look like is Delta X times f of
23
00:01:56.920 --> 00:02:00.810 A:middle L:90%
X, not plus, um, f of x
24
00:02:00.810 --> 00:02:02.230 A:middle L:90%
one. So it's gonna be all those mid points
25
00:02:02.230 --> 00:02:05.909 A:middle L:90%
. So that's going to go all the way until
26
00:02:05.920 --> 00:02:08.400 A:middle L:90%
f of x nine. So when we do that
27
00:02:08.400 --> 00:02:12.599 A:middle L:90%
, it's going to be Delta access 0.1 times.
28
00:02:12.599 --> 00:02:15.550 A:middle L:90%
All those values we can calculate and what we end
29
00:02:15.550 --> 00:02:21.509 A:middle L:90%
up getting is 0.243748 So he said that these answers
30
00:02:21.509 --> 00:02:24.009 A:middle L:90%
are extremely close, which is a great way to
31
00:02:24.009 --> 00:02:31.090 A:middle L:90%
estimate integral. Then, for part C, we
32
00:02:31.090 --> 00:02:37.389 A:middle L:90%
want to do it using the S method. So
33
00:02:37.400 --> 00:02:42.539 A:middle L:90%
what that's gonna look like is S 10 is equal
34
00:02:42.539 --> 00:02:49.750 A:middle L:90%
to Delta X over three times F of zero plus
35
00:02:49.750 --> 00:02:55.139 A:middle L:90%
four times f of 0.1 plus two times f of
36
00:02:55.139 --> 00:02:59.580 A:middle L:90%
0.2, and this is going to keep going on
37
00:02:59.580 --> 00:03:00.960 A:middle L:90%
. It's gonna go 4 to 4 to four to
38
00:03:01.340 --> 00:03:04.629 A:middle L:90%
, so that's going to keep going. And then
39
00:03:04.639 --> 00:03:12.580 A:middle L:90%
it'll end up with to F of 08 plus four
40
00:03:12.590 --> 00:03:19.430 A:middle L:90%
F of 0.9 plus f of one that's going to
41
00:03:19.430 --> 00:03:21.990 A:middle L:90%
give us since we know Del Taxes 0.1, we
42
00:03:21.990 --> 00:03:23.819 A:middle L:90%
look at our values, we calculate them and we
43
00:03:23.819 --> 00:03:30.770 A:middle L:90%
end up getting approximately 0.24 3751 So we see these
44
00:03:30.770 --> 00:03:35.340 A:middle L:90%
are all extremely close in approaching the ultimate value that
45
00:03:35.340 --> 00:03:36.159 A:middle L:90%
we're looking for.