[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.62,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.62,0:00:02.78,Default,,0000,0000,0000,,Now let's do a really\Ninteresting problem.
Dialogue: 0,0:00:02.78,0:00:05.29,Default,,0000,0000,0000,,So I have y equals x,\Nand y is equal to x
Dialogue: 0,0:00:05.29,0:00:07.84,Default,,0000,0000,0000,,squared minus 2x\Nright over here.
Dialogue: 0,0:00:07.84,0:00:09.53,Default,,0000,0000,0000,,And we're going to\Nrotate the region
Dialogue: 0,0:00:09.53,0:00:11.02,Default,,0000,0000,0000,,in between these two functions.
Dialogue: 0,0:00:11.02,0:00:13.28,Default,,0000,0000,0000,,So that's this region\Nright over here.
Dialogue: 0,0:00:13.28,0:00:15.61,Default,,0000,0000,0000,,And we're not going to rotate\Nit just around the x-axis,
Dialogue: 0,0:00:15.61,0:00:19.36,Default,,0000,0000,0000,,we're going to rotate it around\Nthe horizontal line y equals 4.
Dialogue: 0,0:00:19.36,0:00:21.45,Default,,0000,0000,0000,,So we're going to\Nrotate it around this.
Dialogue: 0,0:00:21.45,0:00:23.87,Default,,0000,0000,0000,,And if we do that, we'll get\Na shape that looks like this.
Dialogue: 0,0:00:23.87,0:00:26.55,Default,,0000,0000,0000,,I drew it ahead of time, just\Nso I could draw it nicely.
Dialogue: 0,0:00:26.55,0:00:30.21,Default,,0000,0000,0000,,And as you can see, it looks\Nlike some type of a vase
Dialogue: 0,0:00:30.21,0:00:32.06,Default,,0000,0000,0000,,with a hole at the bottom.
Dialogue: 0,0:00:32.06,0:00:34.52,Default,,0000,0000,0000,,And so what we're going to do\Nis attempt to do this using,
Dialogue: 0,0:00:34.52,0:00:36.40,Default,,0000,0000,0000,,I guess you'd call it\Nthe washer method which
Dialogue: 0,0:00:36.40,0:00:37.80,Default,,0000,0000,0000,,is a variant of the disk method.
Dialogue: 0,0:00:37.80,0:00:39.79,Default,,0000,0000,0000,,So let's construct a washer.
Dialogue: 0,0:00:39.79,0:00:42.35,Default,,0000,0000,0000,,So let's look at a given x.
Dialogue: 0,0:00:42.35,0:00:45.52,Default,,0000,0000,0000,,So let's say an x\Nright over here.
Dialogue: 0,0:00:45.52,0:00:47.75,Default,,0000,0000,0000,,So let's say that we're\Nat an x right over there.
Dialogue: 0,0:00:47.75,0:00:48.37,Default,,0000,0000,0000,,And what we're\Ngoing to do is we're
Dialogue: 0,0:00:48.37,0:00:50.16,Default,,0000,0000,0000,,going to rotate this region.
Dialogue: 0,0:00:50.16,0:00:54.09,Default,,0000,0000,0000,,We're going to give\Nit some depth, dx.
Dialogue: 0,0:00:54.09,0:00:55.01,Default,,0000,0000,0000,,So that is dx.
Dialogue: 0,0:00:55.01,0:00:57.12,Default,,0000,0000,0000,,We're going to rotate\Nthis around the line y
Dialogue: 0,0:00:57.12,0:00:57.84,Default,,0000,0000,0000,,is equal to 4.
Dialogue: 0,0:00:57.84,0:01:02.58,Default,,0000,0000,0000,,So if you were to visualize it\Nover here, you have some depth.
Dialogue: 0,0:01:02.58,0:01:05.47,Default,,0000,0000,0000,,And when you rotate it\Naround, the inner radius
Dialogue: 0,0:01:05.47,0:01:07.68,Default,,0000,0000,0000,,is going to look like the\Ninner radius of our washer.
Dialogue: 0,0:01:07.68,0:01:09.44,Default,,0000,0000,0000,,It's going to look\Nsomething like that.
Dialogue: 0,0:01:09.44,0:01:12.21,Default,,0000,0000,0000,,
Dialogue: 0,0:01:12.21,0:01:13.90,Default,,0000,0000,0000,,And then the outer\Nradius of our washer
Dialogue: 0,0:01:13.90,0:01:17.42,Default,,0000,0000,0000,,is going to contour\Naround x squared minus 2x.
Dialogue: 0,0:01:17.42,0:01:21.78,Default,,0000,0000,0000,,So it's going to\Nlook something--
Dialogue: 0,0:01:21.78,0:01:23.70,Default,,0000,0000,0000,,my best attempt\Nto draw it-- it's
Dialogue: 0,0:01:23.70,0:01:25.44,Default,,0000,0000,0000,,going to look\Nsomething like that.
Dialogue: 0,0:01:25.44,0:01:27.95,Default,,0000,0000,0000,,
Dialogue: 0,0:01:27.95,0:01:30.81,Default,,0000,0000,0000,,And of course, our washer\Nis going to have some depth.
Dialogue: 0,0:01:30.81,0:01:32.25,Default,,0000,0000,0000,,So let me draw the depth.
Dialogue: 0,0:01:32.25,0:01:35.95,Default,,0000,0000,0000,,So it's going to\Nhave some depth, dx.
Dialogue: 0,0:01:35.95,0:01:39.94,Default,,0000,0000,0000,,So this is my best attempt at\Ndrawing some of that the depth.
Dialogue: 0,0:01:39.94,0:01:42.97,Default,,0000,0000,0000,,So this is the\Ndepth of our washer.
Dialogue: 0,0:01:42.97,0:01:45.44,Default,,0000,0000,0000,,And then just to make the face\Nof the washer a little bit
Dialogue: 0,0:01:45.44,0:01:47.23,Default,,0000,0000,0000,,clearer, let me do it\Nin this green color.
Dialogue: 0,0:01:47.23,0:01:49.44,Default,,0000,0000,0000,,So the face of the\Nwasher is going
Dialogue: 0,0:01:49.44,0:01:52.31,Default,,0000,0000,0000,,to be all of this business.
Dialogue: 0,0:01:52.31,0:01:57.06,Default,,0000,0000,0000,,All of this business is going\Nto be the face of our washer.
Dialogue: 0,0:01:57.06,0:01:59.35,Default,,0000,0000,0000,,So if we can figure\Nout the volume of one
Dialogue: 0,0:01:59.35,0:02:01.21,Default,,0000,0000,0000,,of these washers for a\Ngiven x, then we just
Dialogue: 0,0:02:01.21,0:02:03.24,Default,,0000,0000,0000,,have to sum up all of\Nthe washers for all
Dialogue: 0,0:02:03.24,0:02:05.92,Default,,0000,0000,0000,,of the x's in our interval.
Dialogue: 0,0:02:05.92,0:02:07.71,Default,,0000,0000,0000,,So let's see if we can\Nset up the integral,
Dialogue: 0,0:02:07.71,0:02:09.99,Default,,0000,0000,0000,,and maybe in the\Nnext video we'll
Dialogue: 0,0:02:09.99,0:02:13.82,Default,,0000,0000,0000,,just forge ahead and actually\Nevaluate the integral.
Dialogue: 0,0:02:13.82,0:02:15.96,Default,,0000,0000,0000,,So let's think about the\Nvolume of the washer.
Dialogue: 0,0:02:15.96,0:02:17.63,Default,,0000,0000,0000,,To think about the\Nvolume of the washer,
Dialogue: 0,0:02:17.63,0:02:19.58,Default,,0000,0000,0000,,we really just have to\Nthink about the area
Dialogue: 0,0:02:19.58,0:02:21.51,Default,,0000,0000,0000,,of the face of the washer.
Dialogue: 0,0:02:21.51,0:02:26.51,Default,,0000,0000,0000,,So area of "face"--\Nput face in quotes--
Dialogue: 0,0:02:26.51,0:02:28.38,Default,,0000,0000,0000,,is going to be equal to what?
Dialogue: 0,0:02:28.38,0:02:30.67,Default,,0000,0000,0000,,Well, it would be the\Narea of the washer--
Dialogue: 0,0:02:30.67,0:02:32.88,Default,,0000,0000,0000,,if it wasn't a washer,\Nif it was just a coin--
Dialogue: 0,0:02:32.88,0:02:35.44,Default,,0000,0000,0000,,and then subtract out\Nthe area of the part
Dialogue: 0,0:02:35.44,0:02:36.44,Default,,0000,0000,0000,,that you're cutting out.
Dialogue: 0,0:02:36.44,0:02:38.89,Default,,0000,0000,0000,,So the area of the\Nwasher if it didn't
Dialogue: 0,0:02:38.89,0:02:40.75,Default,,0000,0000,0000,,have a hole in the\Nmiddle would just
Dialogue: 0,0:02:40.75,0:02:44.28,Default,,0000,0000,0000,,be pi times the\Nouter radius squared.
Dialogue: 0,0:02:44.28,0:02:48.15,Default,,0000,0000,0000,,
Dialogue: 0,0:02:48.15,0:02:51.09,Default,,0000,0000,0000,,It would be pi times\Nthis radius squared,
Dialogue: 0,0:02:51.09,0:02:52.95,Default,,0000,0000,0000,,that we could call\Nthe outer radius.
Dialogue: 0,0:02:52.95,0:02:55.32,Default,,0000,0000,0000,,And since it's a washer,\Nwe need to subtract out
Dialogue: 0,0:02:55.32,0:02:57.03,Default,,0000,0000,0000,,the area of this inner circle.
Dialogue: 0,0:02:57.03,0:03:05.50,Default,,0000,0000,0000,,So minus pi times\Ninner radius squared.
Dialogue: 0,0:03:05.50,0:03:07.00,Default,,0000,0000,0000,,So we really just\Nhave to figure out
Dialogue: 0,0:03:07.00,0:03:11.16,Default,,0000,0000,0000,,what the outer and inner radius,\Nor radii I should say, are.
Dialogue: 0,0:03:11.16,0:03:12.71,Default,,0000,0000,0000,,So let's think about it.
Dialogue: 0,0:03:12.71,0:03:19.97,Default,,0000,0000,0000,,So our outer radius is\Ngoing to be equal to what?
Dialogue: 0,0:03:19.97,0:03:21.47,Default,,0000,0000,0000,,Well, we can visualize\Nit over here.
Dialogue: 0,0:03:21.47,0:03:23.61,Default,,0000,0000,0000,,This is our outer\Nradius, which is also
Dialogue: 0,0:03:23.61,0:03:27.77,Default,,0000,0000,0000,,going to be equal to\Nthat right over there.
Dialogue: 0,0:03:27.77,0:03:29.72,Default,,0000,0000,0000,,So that's the distance\Nbetween y equals
Dialogue: 0,0:03:29.72,0:03:32.44,Default,,0000,0000,0000,,4 and the function that's\Ndefining our outside.
Dialogue: 0,0:03:32.44,0:03:38.25,Default,,0000,0000,0000,,
Dialogue: 0,0:03:38.25,0:03:40.52,Default,,0000,0000,0000,,So this is essentially,\Nthis height right over here,
Dialogue: 0,0:03:40.52,0:03:45.28,Default,,0000,0000,0000,,is going to be equal to 4\Nminus x squared minus 2x.
Dialogue: 0,0:03:45.28,0:03:48.00,Default,,0000,0000,0000,,I'm just finding the distance\Nor the height between these two
Dialogue: 0,0:03:48.00,0:03:48.90,Default,,0000,0000,0000,,functions.
Dialogue: 0,0:03:48.90,0:03:52.14,Default,,0000,0000,0000,,So the outer radius is\Ngoing to be 4 minus this,
Dialogue: 0,0:03:52.14,0:03:55.10,Default,,0000,0000,0000,,minus x squared minus\N2x, which is just 4
Dialogue: 0,0:03:55.10,0:03:58.52,Default,,0000,0000,0000,,minus x squared plus 2x.
Dialogue: 0,0:03:58.52,0:03:59.85,Default,,0000,0000,0000,,Now, what is the inner radius?
Dialogue: 0,0:03:59.85,0:04:05.08,Default,,0000,0000,0000,,
Dialogue: 0,0:04:05.08,0:04:06.83,Default,,0000,0000,0000,,What is that going to be?
Dialogue: 0,0:04:06.83,0:04:11.55,Default,,0000,0000,0000,,Well, that's just going to\Nbe this distance between y
Dialogue: 0,0:04:11.55,0:04:13.48,Default,,0000,0000,0000,,equals 4 and y equals x.
Dialogue: 0,0:04:13.48,0:04:15.26,Default,,0000,0000,0000,,So it's just going\Nto be 4 minus x.
Dialogue: 0,0:04:15.26,0:04:19.08,Default,,0000,0000,0000,,
Dialogue: 0,0:04:19.08,0:04:22.81,Default,,0000,0000,0000,,So if we wanted to find\Nthe area of the face of one
Dialogue: 0,0:04:22.81,0:04:27.09,Default,,0000,0000,0000,,of these washers for a\Ngiven x, it's going to be--
Dialogue: 0,0:04:27.09,0:04:30.29,Default,,0000,0000,0000,,and we can factor\Nout this pi-- it's
Dialogue: 0,0:04:30.29,0:04:34.68,Default,,0000,0000,0000,,going to be pi times the\Nouter radius squared,
Dialogue: 0,0:04:34.68,0:04:36.54,Default,,0000,0000,0000,,which is all of this\Nbusiness squared.
Dialogue: 0,0:04:36.54,0:04:41.83,Default,,0000,0000,0000,,So it's going to be 4 minus\Nx squared plus 2x squared
Dialogue: 0,0:04:41.83,0:04:43.28,Default,,0000,0000,0000,,minus pi times\Nthe inner radius--
Dialogue: 0,0:04:43.28,0:04:44.78,Default,,0000,0000,0000,,although we factored\Nout the pi-- so
Dialogue: 0,0:04:44.78,0:04:46.81,Default,,0000,0000,0000,,minus the inner radius squared.
Dialogue: 0,0:04:46.81,0:04:51.80,Default,,0000,0000,0000,,So minus 4 minus x squared.
Dialogue: 0,0:04:51.80,0:04:57.65,Default,,0000,0000,0000,,So this will give us\Nthe area of the surface
Dialogue: 0,0:04:57.65,0:04:59.28,Default,,0000,0000,0000,,or the face of one\Nof these washers.
Dialogue: 0,0:04:59.28,0:05:01.70,Default,,0000,0000,0000,,If we want the volume of\None of those watchers,
Dialogue: 0,0:05:01.70,0:05:05.04,Default,,0000,0000,0000,,we then just have to\Nmultiply times the depth, dx.
Dialogue: 0,0:05:05.04,0:05:08.01,Default,,0000,0000,0000,,
Dialogue: 0,0:05:08.01,0:05:10.80,Default,,0000,0000,0000,,And then if we want to actually\Nfind the volume of this entire
Dialogue: 0,0:05:10.80,0:05:14.25,Default,,0000,0000,0000,,figure, then we just have to\Nsum up all of these washers
Dialogue: 0,0:05:14.25,0:05:15.95,Default,,0000,0000,0000,,for each of our x's.
Dialogue: 0,0:05:15.95,0:05:16.83,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:05:16.83,0:05:19.05,Default,,0000,0000,0000,,So we're going to sum\Nup the washers for each
Dialogue: 0,0:05:19.05,0:05:21.30,Default,,0000,0000,0000,,of our x's and take the\Nlimit as they approach zero,
Dialogue: 0,0:05:21.30,0:05:23.49,Default,,0000,0000,0000,,but we have to make sure\Nwe got our interval right.
Dialogue: 0,0:05:23.49,0:05:26.26,Default,,0000,0000,0000,,So what are these-- we care\Nabout the entire region
Dialogue: 0,0:05:26.26,0:05:28.84,Default,,0000,0000,0000,,between the points\Nwhere they intersect.
Dialogue: 0,0:05:28.84,0:05:30.62,Default,,0000,0000,0000,,So let's make sure\Nwe get our interval.
Dialogue: 0,0:05:30.62,0:05:32.20,Default,,0000,0000,0000,,So to figure out our\Ninterval, we just
Dialogue: 0,0:05:32.20,0:05:36.07,Default,,0000,0000,0000,,say when does y equal\Nx intersect y equal
Dialogue: 0,0:05:36.07,0:05:37.35,Default,,0000,0000,0000,,x squared minus 2x?
Dialogue: 0,0:05:37.35,0:05:39.99,Default,,0000,0000,0000,,
Dialogue: 0,0:05:39.99,0:05:41.63,Default,,0000,0000,0000,,Let me do this in\Na different color.
Dialogue: 0,0:05:41.63,0:05:44.15,Default,,0000,0000,0000,,We just have to\Nthink about when does
Dialogue: 0,0:05:44.15,0:05:46.24,Default,,0000,0000,0000,,x equal x squared minus 2x.
Dialogue: 0,0:05:46.24,0:05:49.37,Default,,0000,0000,0000,,
Dialogue: 0,0:05:49.37,0:05:51.42,Default,,0000,0000,0000,,When are our two functions\Nequal to each other?
Dialogue: 0,0:05:51.42,0:05:53.03,Default,,0000,0000,0000,,Which is equivalent\Nto-- if we just
Dialogue: 0,0:05:53.03,0:05:58.80,Default,,0000,0000,0000,,subtract x from\Nboth sides, we get
Dialogue: 0,0:05:58.80,0:06:02.19,Default,,0000,0000,0000,,when does x squared\Nminus 3x equal 0.
Dialogue: 0,0:06:02.19,0:06:05.26,Default,,0000,0000,0000,,We can factor out an x\Non the right hand side.
Dialogue: 0,0:06:05.26,0:06:09.53,Default,,0000,0000,0000,,So this is going to be when does\Nx times x minus 3 equal zero.
Dialogue: 0,0:06:09.53,0:06:12.31,Default,,0000,0000,0000,,Well, if the product is equal\Nto 0, at least one of these
Dialogue: 0,0:06:12.31,0:06:13.35,Default,,0000,0000,0000,,need to be equal to 0.
Dialogue: 0,0:06:13.35,0:06:18.38,Default,,0000,0000,0000,,So x could be equal to 0,\Nor x minus 3 is equal to 0.
Dialogue: 0,0:06:18.38,0:06:21.28,Default,,0000,0000,0000,,So x is equal to 0\Nor x is equal to 3.
Dialogue: 0,0:06:21.28,0:06:24.01,Default,,0000,0000,0000,,So this is x is 0, and\Nthis right over here
Dialogue: 0,0:06:24.01,0:06:25.85,Default,,0000,0000,0000,,is x is equal to 3.
Dialogue: 0,0:06:25.85,0:06:27.10,Default,,0000,0000,0000,,So that gives us our interval.
Dialogue: 0,0:06:27.10,0:06:29.12,Default,,0000,0000,0000,,We're going to go\Nfrom x equals 0
Dialogue: 0,0:06:29.12,0:06:32.85,Default,,0000,0000,0000,,to x equals 3 to get our volume.
Dialogue: 0,0:06:32.85,0:06:35.00,Default,,0000,0000,0000,,In the next video,\Nwe'll actually
Dialogue: 0,0:06:35.00,0:06:37.32,Default,,0000,0000,0000,,evaluate this integral.