WEBVTT
1
00:00:00.240 --> 00:00:03.589 A:middle L:90%
So were given a solid S. And we're asked
2
00:00:03.589 --> 00:00:07.459 A:middle L:90%
to find the volume of this solid S. Is
3
00:00:07.940 --> 00:00:13.910 A:middle L:90%
a cap of a sphere bon Jovi. Yeah with
4
00:00:13.910 --> 00:00:20.980 A:middle L:90%
the radius R. Wilson and a height. Uh
5
00:00:20.980 --> 00:00:29.350 A:middle L:90%
huh. Each there's some like and deep in yes
6
00:00:32.140 --> 00:00:36.049 A:middle L:90%
. Now in order to um find the volume of
7
00:00:36.049 --> 00:00:41.130 A:middle L:90%
this cap, we'll draw a circle however we'll draw
8
00:00:41.130 --> 00:00:45.750 A:middle L:90%
it in a certain way. So set up our
9
00:00:45.750 --> 00:00:47.929 A:middle L:90%
X. And Y axes and then we'll have a
10
00:00:47.929 --> 00:00:53.719 A:middle L:90%
circle like this. The distance from the origin to
11
00:00:53.719 --> 00:00:57.159 A:middle L:90%
this point on the right is going to be the
12
00:00:57.159 --> 00:01:03.859 A:middle L:90%
height of the cap H. And so God works
13
00:01:04.989 --> 00:01:07.760 A:middle L:90%
. The radius of the circle is going to be
14
00:01:08.140 --> 00:01:15.590 A:middle L:90%
Yeah. Yeah. Words. Yeah. A little
15
00:01:15.590 --> 00:01:19.939 A:middle L:90%
r We got the pork scores, murdering sucks.
16
00:01:21.739 --> 00:01:26.049 A:middle L:90%
Yeah. All right. No I didn't even finish
17
00:01:26.840 --> 00:01:30.730 A:middle L:90%
. But you said you're really now since the whole
18
00:01:30.730 --> 00:01:34.849 A:middle L:90%
radius is our follows at the center of this circle
19
00:01:34.849 --> 00:01:41.510 A:middle L:90%
is at the point let's see this is our,
20
00:01:41.519 --> 00:01:47.319 A:middle L:90%
sorry negative are plus H. Zero. You know
21
00:01:47.319 --> 00:01:49.590 A:middle L:90%
c. three. And therefore the equation for a
22
00:01:49.590 --> 00:01:55.760 A:middle L:90%
circle is X. That's kind of what I wanted
23
00:01:56.640 --> 00:02:04.079 A:middle L:90%
minus H minus R squared plus Y squared equals the
24
00:02:04.079 --> 00:02:10.280 A:middle L:90%
radius R squared those better what? Look you can't
25
00:02:10.280 --> 00:02:16.139 A:middle L:90%
say hard ours. Here is your anger. Anger
26
00:02:16.139 --> 00:02:20.219 A:middle L:90%
reached a hard art. Then they got James,
27
00:02:20.219 --> 00:02:23.159 A:middle L:90%
Earl jones to play Darth vader because he's black blue
28
00:02:23.639 --> 00:02:30.849 A:middle L:90%
. I am now imagine rotating this across the X
29
00:02:30.849 --> 00:02:38.750 A:middle L:90%
axis. But we end up with is our cap
30
00:02:38.139 --> 00:02:44.789 A:middle L:90%
and we end up with disks use discs have a
31
00:02:44.789 --> 00:02:52.689 A:middle L:90%
radius which is why equals the positive square root of
32
00:02:52.699 --> 00:02:58.250 A:middle L:90%
r squared minus x minus h minus r squared.
33
00:02:58.840 --> 00:03:06.539 A:middle L:90%
It was it was and shylock. So great they
34
00:03:06.539 --> 00:03:08.939 A:middle L:90%
would play console. This is one way to approach
35
00:03:08.939 --> 00:03:14.169 A:middle L:90%
it. I never noticed, let's say. And
36
00:03:14.169 --> 00:03:16.379 A:middle L:90%
so the volume V by the disk method is given
37
00:03:16.379 --> 00:03:23.800 A:middle L:90%
by the pi times the integral from x equals 02
38
00:03:23.810 --> 00:03:25.629 A:middle L:90%
H. Of our radius which is the square root
39
00:03:25.629 --> 00:03:31.879 A:middle L:90%
of r squared minus x minus h minus R squared
40
00:03:31.889 --> 00:03:39.379 A:middle L:90%
all squared dx one. So simplifying this is pi
41
00:03:39.379 --> 00:03:43.639 A:middle L:90%
times the integral from zero to H. Of r
42
00:03:43.639 --> 00:03:52.750 A:middle L:90%
squared minus x minus h minus or squared dx Mhm
43
00:03:53.639 --> 00:03:57.680 A:middle L:90%
. If you take anti derivatives you should get pi
44
00:03:57.680 --> 00:04:06.310 A:middle L:90%
times this is r squared x minus and then one
45
00:04:06.310 --> 00:04:14.599 A:middle L:90%
third times x minus H minus R cubed From x
46
00:04:14.599 --> 00:04:25.360 A:middle L:90%
equals zero two H. Plugging in. We get
47
00:04:25.740 --> 00:04:35.660 A:middle L:90%
pi times R squared h-1 3rd of yeah are
48
00:04:38.939 --> 00:04:46.759 A:middle L:90%
understood- and this is 0-1/3 of. Sorry
49
00:04:46.759 --> 00:04:50.680 A:middle L:90%
there should be r cubed. My mistake I as
50
00:04:50.680 --> 00:04:56.730 A:middle L:90%
a third times negative H minus R cubed which is
51
00:04:56.730 --> 00:05:02.420 A:middle L:90%
plus one third times h minus r cubed rider bikes
52
00:05:02.629 --> 00:05:05.100 A:middle L:90%
at each other and then try to joe get him
53
00:05:05.110 --> 00:05:09.410 A:middle L:90%
through the spokes data, lock the bike up,
54
00:05:09.420 --> 00:05:13.579 A:middle L:90%
you fucking fall off. That's what and so simplifying
55
00:05:13.579 --> 00:05:17.160 A:middle L:90%
. This is pi times R squared h minus one
56
00:05:17.160 --> 00:05:21.670 A:middle L:90%
third R cubed. And this is actually plus 1/3
57
00:05:21.670 --> 00:05:28.939 A:middle L:90%
times ar- H cubed. Yeah you could leave
58
00:05:28.939 --> 00:05:36.350 A:middle L:90%
in this form or simplify further so mhm slip behind
59
00:05:38.040 --> 00:05:41.259 A:middle L:90%
in the bars and then got stuck. If you
60
00:05:41.269 --> 00:05:47.899 A:middle L:90%
multiply everything out this will eventually simplify too. One
61
00:05:47.899 --> 00:05:55.480 A:middle L:90%
third times kids are spear liss. We factor that
62
00:05:55.480 --> 00:05:59.120 A:middle L:90%
one third pi times three R squared H minus R
63
00:05:59.120 --> 00:06:03.139 A:middle L:90%
cubed plus ar minus H cubed. And then we
64
00:06:03.139 --> 00:06:10.089 A:middle L:90%
get one third pie and after several steps three R
65
00:06:10.100 --> 00:06:15.089 A:middle L:90%
h squared minus H cubed which is equal to one
66
00:06:15.089 --> 00:06:23.610 A:middle L:90%
third pie H squared Times three AR- H dump
67
00:06:23.610 --> 00:06:28.069 A:middle L:90%
every girl that gets her period and disrespect going to
68
00:06:28.069 --> 00:06:30.060 A:middle L:90%
come. You gonna try in the period with me
69
00:06:30.439 --> A:middle L:90%
? It's