0:00:00.620,0:00:02.780
Now let's do a really[br]interesting problem.

0:00:02.780,0:00:05.290
So I have y equals x,[br]and y is equal to x

0:00:05.290,0:00:07.840
squared minus 2x[br]right over here.

0:00:07.840,0:00:09.530
And we're going to[br]rotate the region

0:00:09.530,0:00:11.020
in between these two functions.

0:00:11.020,0:00:13.277
So that's this region[br]right over here.

0:00:13.277,0:00:15.610
And we're not going to rotate[br]it just around the x-axis,

0:00:15.610,0:00:19.360
we're going to rotate it around[br]the horizontal line y equals 4.

0:00:19.360,0:00:21.454
So we're going to[br]rotate it around this.

0:00:21.454,0:00:23.870
And if we do that, we'll get[br]a shape that looks like this.

0:00:23.870,0:00:26.550
I drew it ahead of time, just[br]so I could draw it nicely.

0:00:26.550,0:00:30.210
And as you can see, it looks[br]like some type of a vase

0:00:30.210,0:00:32.060
with a hole at the bottom.

0:00:32.060,0:00:34.520
And so what we're going to do[br]is attempt to do this using,

0:00:34.520,0:00:36.395
I guess you'd call it[br]the washer method which

0:00:36.395,0:00:37.800
is a variant of the disk method.

0:00:37.800,0:00:39.790
So let's construct a washer.

0:00:39.790,0:00:42.350
So let's look at a given x.

0:00:42.350,0:00:45.520
So let's say an x[br]right over here.

0:00:45.520,0:00:47.750
So let's say that we're[br]at an x right over there.

0:00:47.750,0:00:48.370
And what we're[br]going to do is we're

0:00:48.370,0:00:50.160
going to rotate this region.

0:00:50.160,0:00:54.090
We're going to give[br]it some depth, dx.

0:00:54.090,0:00:55.010
So that is dx.

0:00:55.010,0:00:57.120
We're going to rotate[br]this around the line y

0:00:57.120,0:00:57.840
is equal to 4.

0:00:57.840,0:01:02.580
So if you were to visualize it[br]over here, you have some depth.

0:01:02.580,0:01:05.472
And when you rotate it[br]around, the inner radius

0:01:05.472,0:01:07.680
is going to look like the[br]inner radius of our washer.

0:01:07.680,0:01:09.440
It's going to look[br]something like that.

0:01:12.210,0:01:13.900
And then the outer[br]radius of our washer

0:01:13.900,0:01:17.420
is going to contour[br]around x squared minus 2x.

0:01:17.420,0:01:21.780
So it's going to[br]look something--

0:01:21.780,0:01:23.703
my best attempt[br]to draw it-- it's

0:01:23.703,0:01:25.440
going to look[br]something like that.

0:01:27.950,0:01:30.810
And of course, our washer[br]is going to have some depth.

0:01:30.810,0:01:32.250
So let me draw the depth.

0:01:32.250,0:01:35.950
So it's going to[br]have some depth, dx.

0:01:35.950,0:01:39.940
So this is my best attempt at[br]drawing some of that the depth.

0:01:39.940,0:01:42.970
So this is the[br]depth of our washer.

0:01:42.970,0:01:45.440
And then just to make the face[br]of the washer a little bit

0:01:45.440,0:01:47.230
clearer, let me do it[br]in this green color.

0:01:47.230,0:01:49.440
So the face of the[br]washer is going

0:01:49.440,0:01:52.310
to be all of this business.

0:01:52.310,0:01:57.060
All of this business is going[br]to be the face of our washer.

0:01:57.060,0:01:59.350
So if we can figure[br]out the volume of one

0:01:59.350,0:02:01.210
of these washers for a[br]given x, then we just

0:02:01.210,0:02:03.240
have to sum up all of[br]the washers for all

0:02:03.240,0:02:05.919
of the x's in our interval.

0:02:05.919,0:02:07.710
So let's see if we can[br]set up the integral,

0:02:07.710,0:02:09.990
and maybe in the[br]next video we'll

0:02:09.990,0:02:13.820
just forge ahead and actually[br]evaluate the integral.

0:02:13.820,0:02:15.964
So let's think about the[br]volume of the washer.

0:02:15.964,0:02:17.630
To think about the[br]volume of the washer,

0:02:17.630,0:02:19.580
we really just have to[br]think about the area

0:02:19.580,0:02:21.510
of the face of the washer.

0:02:21.510,0:02:26.510
So area of "face"--[br]put face in quotes--

0:02:26.510,0:02:28.380
is going to be equal to what?

0:02:28.380,0:02:30.670
Well, it would be the[br]area of the washer--

0:02:30.670,0:02:32.880
if it wasn't a washer,[br]if it was just a coin--

0:02:32.880,0:02:35.441
and then subtract out[br]the area of the part

0:02:35.441,0:02:36.440
that you're cutting out.

0:02:36.440,0:02:38.890
So the area of the[br]washer if it didn't

0:02:38.890,0:02:40.750
have a hole in the[br]middle would just

0:02:40.750,0:02:44.285
be pi times the[br]outer radius squared.

0:02:48.150,0:02:51.090
It would be pi times[br]this radius squared,

0:02:51.090,0:02:52.950
that we could call[br]the outer radius.

0:02:52.950,0:02:55.320
And since it's a washer,[br]we need to subtract out

0:02:55.320,0:02:57.030
the area of this inner circle.

0:02:57.030,0:03:05.500
So minus pi times[br]inner radius squared.

0:03:05.500,0:03:07.000
So we really just[br]have to figure out

0:03:07.000,0:03:11.160
what the outer and inner radius,[br]or radii I should say, are.

0:03:11.160,0:03:12.710
So let's think about it.

0:03:12.710,0:03:19.971
So our outer radius is[br]going to be equal to what?

0:03:19.971,0:03:21.470
Well, we can visualize[br]it over here.

0:03:21.470,0:03:23.610
This is our outer[br]radius, which is also

0:03:23.610,0:03:27.770
going to be equal to[br]that right over there.

0:03:27.770,0:03:29.720
So that's the distance[br]between y equals

0:03:29.720,0:03:32.440
4 and the function that's[br]defining our outside.

0:03:38.250,0:03:40.520
So this is essentially,[br]this height right over here,

0:03:40.520,0:03:45.280
is going to be equal to 4[br]minus x squared minus 2x.

0:03:45.280,0:03:48.000
I'm just finding the distance[br]or the height between these two

0:03:48.000,0:03:48.900
functions.

0:03:48.900,0:03:52.140
So the outer radius is[br]going to be 4 minus this,

0:03:52.140,0:03:55.100
minus x squared minus[br]2x, which is just 4

0:03:55.100,0:03:58.520
minus x squared plus 2x.

0:03:58.520,0:03:59.850
Now, what is the inner radius?

0:04:05.080,0:04:06.830
What is that going to be?

0:04:06.830,0:04:11.550
Well, that's just going to[br]be this distance between y

0:04:11.550,0:04:13.480
equals 4 and y equals x.

0:04:13.480,0:04:15.265
So it's just going[br]to be 4 minus x.

0:04:19.079,0:04:22.810
So if we wanted to find[br]the area of the face of one

0:04:22.810,0:04:27.090
of these washers for a[br]given x, it's going to be--

0:04:27.090,0:04:30.290
and we can factor[br]out this pi-- it's

0:04:30.290,0:04:34.680
going to be pi times the[br]outer radius squared,

0:04:34.680,0:04:36.540
which is all of this[br]business squared.

0:04:36.540,0:04:41.830
So it's going to be 4 minus[br]x squared plus 2x squared

0:04:41.830,0:04:43.280
minus pi times[br]the inner radius--

0:04:43.280,0:04:44.780
although we factored[br]out the pi-- so

0:04:44.780,0:04:46.810
minus the inner radius squared.

0:04:46.810,0:04:51.800
So minus 4 minus x squared.

0:04:51.800,0:04:57.650
So this will give us[br]the area of the surface

0:04:57.650,0:04:59.280
or the face of one[br]of these washers.

0:04:59.280,0:05:01.700
If we want the volume of[br]one of those watchers,

0:05:01.700,0:05:05.040
we then just have to[br]multiply times the depth, dx.

0:05:08.010,0:05:10.800
And then if we want to actually[br]find the volume of this entire

0:05:10.800,0:05:14.250
figure, then we just have to[br]sum up all of these washers

0:05:14.250,0:05:15.950
for each of our x's.

0:05:15.950,0:05:16.830
So let's do that.

0:05:16.830,0:05:19.050
So we're going to sum[br]up the washers for each

0:05:19.050,0:05:21.300
of our x's and take the[br]limit as they approach zero,

0:05:21.300,0:05:23.490
but we have to make sure[br]we got our interval right.

0:05:23.490,0:05:26.260
So what are these-- we care[br]about the entire region

0:05:26.260,0:05:28.840
between the points[br]where they intersect.

0:05:28.840,0:05:30.620
So let's make sure[br]we get our interval.

0:05:30.620,0:05:32.203
So to figure out our[br]interval, we just

0:05:32.203,0:05:36.070
say when does y equal[br]x intersect y equal

0:05:36.070,0:05:37.350
x squared minus 2x?

0:05:39.990,0:05:41.630
Let me do this in[br]a different color.

0:05:41.630,0:05:44.150
We just have to[br]think about when does

0:05:44.150,0:05:46.245
x equal x squared minus 2x.

0:05:49.370,0:05:51.420
When are our two functions[br]equal to each other?

0:05:51.420,0:05:53.030
Which is equivalent[br]to-- if we just

0:05:53.030,0:05:58.800
subtract x from[br]both sides, we get

0:05:58.800,0:06:02.190
when does x squared[br]minus 3x equal 0.

0:06:02.190,0:06:05.260
We can factor out an x[br]on the right hand side.

0:06:05.260,0:06:09.530
So this is going to be when does[br]x times x minus 3 equal zero.

0:06:09.530,0:06:12.310
Well, if the product is equal[br]to 0, at least one of these

0:06:12.310,0:06:13.350
need to be equal to 0.

0:06:13.350,0:06:18.380
So x could be equal to 0,[br]or x minus 3 is equal to 0.

0:06:18.380,0:06:21.280
So x is equal to 0[br]or x is equal to 3.

0:06:21.280,0:06:24.010
So this is x is 0, and[br]this right over here

0:06:24.010,0:06:25.850
is x is equal to 3.

0:06:25.850,0:06:27.100
So that gives us our interval.

0:06:27.100,0:06:29.120
We're going to go[br]from x equals 0

0:06:29.120,0:06:32.850
to x equals 3 to get our volume.

0:06:32.850,0:06:35.000
In the next video,[br]we'll actually

0:06:35.000,0:06:37.320
evaluate this integral.