WEBVTT
1
00:00:01.639 --> 00:00:05.269 A:middle L:90%
All right, let's go ahead and answer the question
2
00:00:06.639 --> 00:00:13.509 A:middle L:90%
. The integral from negative 1 to 2 off,
3
00:00:13.519 --> 00:00:19.769 A:middle L:90%
one minus x dx. And then the question specifically
4
00:00:19.769 --> 00:00:24.350 A:middle L:90%
asks you to evaluate using in terms off the geometrical
5
00:00:24.350 --> 00:00:29.940 A:middle L:90%
approach. Okay, What does that mean? Basically
6
00:00:29.940 --> 00:00:33.200 A:middle L:90%
, there are rules to integration that you can utilize
7
00:00:33.200 --> 00:00:36.799 A:middle L:90%
to solve the problem analytically. But remember, earthy
8
00:00:36.810 --> 00:00:40.119 A:middle L:90%
, integral can be conceptually thought, often area under
9
00:00:40.119 --> 00:00:43.539 A:middle L:90%
the curve. So if you know how to calculate
10
00:00:43.539 --> 00:00:46.990 A:middle L:90%
that area geometrically, you could also approach it that
11
00:00:47.000 --> 00:00:51.890 A:middle L:90%
way. So let's analyze this on the X Y
12
00:00:51.890 --> 00:00:54.299 A:middle L:90%
coordinate system. We are going to be able to
13
00:00:54.310 --> 00:00:57.259 A:middle L:90%
graph this thing quite precisely. It's one of the
14
00:00:57.259 --> 00:01:00.670 A:middle L:90%
first things that you learned in algebra. One minus
15
00:01:00.679 --> 00:01:03.920 A:middle L:90%
X has three ey intercept that one. X intercept
16
00:01:03.920 --> 00:01:07.540 A:middle L:90%
is also one because we know that the slope is
17
00:01:07.540 --> 00:01:08.900 A:middle L:90%
negative. One. Okay, see if they can
18
00:01:08.900 --> 00:01:15.650 A:middle L:90%
draw a nice street line. There you go.
19
00:01:15.239 --> 00:01:19.269 A:middle L:90%
The limit of integration is from negative. Wanted to
20
00:01:19.280 --> 00:01:22.730 A:middle L:90%
so negative one is right there. Two is right
21
00:01:22.730 --> 00:01:26.129 A:middle L:90%
here. So what is the area under the curve
22
00:01:26.140 --> 00:01:29.609 A:middle L:90%
? Well, it starts right here, and it
23
00:01:29.620 --> 00:01:33.049 A:middle L:90%
ends over there so you can see that there are
24
00:01:33.049 --> 00:01:36.989 A:middle L:90%
two wrecked ing two triangles right there and you can
25
00:01:36.989 --> 00:01:40.040 A:middle L:90%
see this one is above the X axis. So
26
00:01:40.040 --> 00:01:42.230 A:middle L:90%
it will be a positive area and this one is
27
00:01:42.230 --> 00:01:45.379 A:middle L:90%
below the X axis. So it's going to be
28
00:01:45.379 --> 00:01:48.650 A:middle L:90%
a negative area. You can also see right away
29
00:01:48.650 --> 00:01:52.150 A:middle L:90%
that the red portion is larger than the blue portion
30
00:01:52.159 --> 00:01:56.079 A:middle L:90%
. So the net area under the curve is going
31
00:01:56.079 --> 00:01:57.829 A:middle L:90%
to be a positive number. So we immediately see
32
00:01:57.829 --> 00:02:00.250 A:middle L:90%
that it's going to be a positive answer, okay
33
00:02:00.840 --> 00:02:04.870 A:middle L:90%
? And these air triangles, which are very simple
34
00:02:04.870 --> 00:02:07.729 A:middle L:90%
to calculate We've known this since maybe third or fourth
35
00:02:07.729 --> 00:02:12.430 A:middle L:90%
grade. So that's calculated. The height here is
36
00:02:12.430 --> 00:02:15.759 A:middle L:90%
going to be too. The distance from negative 1
37
00:02:15.759 --> 00:02:16.580 A:middle L:90%
to 1 is also too. So we know that
38
00:02:16.580 --> 00:02:20.479 A:middle L:90%
the area of the red portion is for divided by
39
00:02:20.490 --> 00:02:23.250 A:middle L:90%
two. Just took the blue region. You know
40
00:02:23.250 --> 00:02:25.960 A:middle L:90%
, that wanted to is one the height here is
41
00:02:25.960 --> 00:02:30.569 A:middle L:90%
one. So one divided by two is one half
42
00:02:30.539 --> 00:02:35.860 A:middle L:90%
. And remember that this is a negative area because
43
00:02:35.860 --> 00:02:38.750 A:middle L:90%
it's below the X axis. Okay, so the
44
00:02:38.750 --> 00:02:40.030 A:middle L:90%
answer is going to be two minus one half,
45
00:02:40.030 --> 00:02:45.050 A:middle L:90%
which is three over to. And that's how you
46
00:02:45.050 --> 00:02:46.159 A:middle L:90%
answer this question.