[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.74,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.74,0:00:04.16,Default,,0000,0000,0000,,Let's say I wanted to find the\Nvolume of a cube, where the
Dialogue: 0,0:00:04.16,0:00:07.15,Default,,0000,0000,0000,,values of the cube-- let's say\Nx is between-- x is greater
Dialogue: 0,0:00:07.15,0:00:10.35,Default,,0000,0000,0000,,than or equal to 0, is less\Nthan or equal to,
Dialogue: 0,0:00:10.35,0:00:11.78,Default,,0000,0000,0000,,I don't know, 3.
Dialogue: 0,0:00:11.78,0:00:14.60,Default,,0000,0000,0000,,Let's say y is greater than\Nor equal to 0, and is
Dialogue: 0,0:00:14.60,0:00:17.00,Default,,0000,0000,0000,,less than or equal to 4.
Dialogue: 0,0:00:17.00,0:00:21.27,Default,,0000,0000,0000,,And then let's say that z is\Ngreater than or equal to 0 and
Dialogue: 0,0:00:21.27,0:00:23.06,Default,,0000,0000,0000,,is less than or equal to 2.
Dialogue: 0,0:00:23.06,0:00:26.65,Default,,0000,0000,0000,,And I know, using basic\Ngeometry you could figure out--
Dialogue: 0,0:00:26.65,0:00:30.37,Default,,0000,0000,0000,,you know, just multiply the\Nwidth times the height times
Dialogue: 0,0:00:30.37,0:00:31.34,Default,,0000,0000,0000,,the depth and you'd\Nhave the volume.
Dialogue: 0,0:00:31.34,0:00:34.28,Default,,0000,0000,0000,,But I want to do this example,\Njust so that you get used to
Dialogue: 0,0:00:34.28,0:00:36.70,Default,,0000,0000,0000,,what a triple integral looks\Nlike, how it relates to a
Dialogue: 0,0:00:36.70,0:00:39.18,Default,,0000,0000,0000,,double integral, and then later\Nin the next video we could do
Dialogue: 0,0:00:39.18,0:00:40.29,Default,,0000,0000,0000,,something slightly\Nmore complicated.
Dialogue: 0,0:00:40.29,0:00:44.04,Default,,0000,0000,0000,,So let's just draw\Nthat, this volume.
Dialogue: 0,0:00:44.04,0:00:51.78,Default,,0000,0000,0000,,So this is my x-axis, this is\Nmy z-axis, this is the y.
Dialogue: 0,0:00:51.78,0:00:54.33,Default,,0000,0000,0000,,
Dialogue: 0,0:00:54.33,0:00:55.80,Default,,0000,0000,0000,,x, y, z.
Dialogue: 0,0:00:55.80,0:00:59.60,Default,,0000,0000,0000,,
Dialogue: 0,0:00:59.60,0:01:00.08,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:01:00.08,0:01:01.91,Default,,0000,0000,0000,,So x is between 0 and 3.
Dialogue: 0,0:01:01.91,0:01:03.07,Default,,0000,0000,0000,,So that's x is equal to 0.
Dialogue: 0,0:01:03.07,0:01:09.12,Default,,0000,0000,0000,,This is x is equal to--\Nlet's see, 1, 2, 3.
Dialogue: 0,0:01:09.12,0:01:10.57,Default,,0000,0000,0000,,y is between 0 and 4.
Dialogue: 0,0:01:10.57,0:01:13.18,Default,,0000,0000,0000,,1, 2, 3, 4.
Dialogue: 0,0:01:13.18,0:01:15.45,Default,,0000,0000,0000,,So the x-y plane will look\Nsomething like this.
Dialogue: 0,0:01:15.45,0:01:20.52,Default,,0000,0000,0000,,The kind of base of our cube\Nwill look something like this.
Dialogue: 0,0:01:20.52,0:01:21.77,Default,,0000,0000,0000,,And then z is between 0 and 2.
Dialogue: 0,0:01:21.77,0:01:25.35,Default,,0000,0000,0000,,So 0 is the x-y plane,\Nand then 1, 2.
Dialogue: 0,0:01:25.35,0:01:27.13,Default,,0000,0000,0000,,So this would be the top part.
Dialogue: 0,0:01:27.13,0:01:30.60,Default,,0000,0000,0000,,And maybe I'll do that in a\Nslightly different color.
Dialogue: 0,0:01:30.60,0:01:34.52,Default,,0000,0000,0000,,So this is along the x-z axis.
Dialogue: 0,0:01:34.52,0:01:36.36,Default,,0000,0000,0000,,You'd have a boundary\Nhere, and then it would
Dialogue: 0,0:01:36.36,0:01:38.32,Default,,0000,0000,0000,,come in like this.
Dialogue: 0,0:01:38.32,0:01:41.85,Default,,0000,0000,0000,,You have a boundary here,\Ncome in like that.
Dialogue: 0,0:01:41.85,0:01:43.81,Default,,0000,0000,0000,,A boundary there.
Dialogue: 0,0:01:43.81,0:01:45.60,Default,,0000,0000,0000,,So we want to figure out\Nthe volume of this cube.
Dialogue: 0,0:01:45.60,0:01:46.37,Default,,0000,0000,0000,,And you could do it.
Dialogue: 0,0:01:46.37,0:01:51.54,Default,,0000,0000,0000,,You could say, well, the depth\Nis 3, the base, the width is 4,
Dialogue: 0,0:01:51.54,0:01:53.92,Default,,0000,0000,0000,,so this area is 12\Ntimes the height.
Dialogue: 0,0:01:53.92,0:01:55.17,Default,,0000,0000,0000,,12 times 2 is 24.
Dialogue: 0,0:01:55.17,0:01:58.98,Default,,0000,0000,0000,,You could say it's 24\Ncubic units, whatever
Dialogue: 0,0:01:58.98,0:01:59.63,Default,,0000,0000,0000,,units we're doing.
Dialogue: 0,0:01:59.63,0:02:01.99,Default,,0000,0000,0000,,But let's do it as\Na triple integral.
Dialogue: 0,0:02:01.99,0:02:03.64,Default,,0000,0000,0000,,So what does a triple\Nintegral mean?
Dialogue: 0,0:02:03.64,0:02:07.11,Default,,0000,0000,0000,,Well, what we could do is we\Ncould take the volume of a very
Dialogue: 0,0:02:07.11,0:02:10.67,Default,,0000,0000,0000,,small-- I don't want to say\Narea-- of a very small volume.
Dialogue: 0,0:02:10.67,0:02:14.77,Default,,0000,0000,0000,,So let's say I wanted to take\Nthe volume of a small cube.
Dialogue: 0,0:02:14.77,0:02:17.81,Default,,0000,0000,0000,,Some place in this-- in the\Nvolume under question.
Dialogue: 0,0:02:17.81,0:02:20.16,Default,,0000,0000,0000,,And it'll start to make more\Nsense, or it starts to become a
Dialogue: 0,0:02:20.16,0:02:22.86,Default,,0000,0000,0000,,lot more useful, when we have\Nvariable boundaries and
Dialogue: 0,0:02:22.86,0:02:25.05,Default,,0000,0000,0000,,surfaces and curves\Nas boundaries.
Dialogue: 0,0:02:25.05,0:02:26.84,Default,,0000,0000,0000,,But let's say we want to\Nfigure out the volume of this
Dialogue: 0,0:02:26.84,0:02:29.78,Default,,0000,0000,0000,,little, small cube here.
Dialogue: 0,0:02:29.78,0:02:30.59,Default,,0000,0000,0000,,That's my cube.
Dialogue: 0,0:02:30.59,0:02:33.63,Default,,0000,0000,0000,,It's some place in this larger\Ncube, this larger rectangle,
Dialogue: 0,0:02:33.63,0:02:35.46,Default,,0000,0000,0000,,cubic rectangle, whatever\Nyou want to call it.
Dialogue: 0,0:02:35.46,0:02:36.54,Default,,0000,0000,0000,,So what's the volume\Nof that cube?
Dialogue: 0,0:02:36.54,0:02:38.93,Default,,0000,0000,0000,,Let's say that its width is dy.
Dialogue: 0,0:02:38.93,0:02:42.32,Default,,0000,0000,0000,,
Dialogue: 0,0:02:42.32,0:02:44.01,Default,,0000,0000,0000,,So that length\Nright there is dy.
Dialogue: 0,0:02:44.01,0:02:46.81,Default,,0000,0000,0000,,It's height is dx.
Dialogue: 0,0:02:46.81,0:02:49.66,Default,,0000,0000,0000,,Sorry, no, it's\Nheight is dz, right?
Dialogue: 0,0:02:49.66,0:02:51.84,Default,,0000,0000,0000,,The way I drew it,\Nz is up and down.
Dialogue: 0,0:02:51.84,0:02:53.86,Default,,0000,0000,0000,,And it's depth is dx.
Dialogue: 0,0:02:53.86,0:02:55.94,Default,,0000,0000,0000,,This is dx.
Dialogue: 0,0:02:55.94,0:02:56.75,Default,,0000,0000,0000,,This is dz.
Dialogue: 0,0:02:56.75,0:02:57.72,Default,,0000,0000,0000,,This is dy.
Dialogue: 0,0:02:57.72,0:03:01.26,Default,,0000,0000,0000,,So you can say that a small\Nvolume within this larger
Dialogue: 0,0:03:01.26,0:03:04.83,Default,,0000,0000,0000,,volume-- you could call that\Ndv, which is kind of the
Dialogue: 0,0:03:04.83,0:03:06.75,Default,,0000,0000,0000,,volume differential.
Dialogue: 0,0:03:06.75,0:03:10.29,Default,,0000,0000,0000,,And that would be equal to,\Nyou could say, it's just
Dialogue: 0,0:03:10.29,0:03:13.99,Default,,0000,0000,0000,,the width times the\Nlength times the height.
Dialogue: 0,0:03:13.99,0:03:15.95,Default,,0000,0000,0000,,dx times dy times dz.
Dialogue: 0,0:03:15.95,0:03:17.76,Default,,0000,0000,0000,,And you could switch the\Norders of these, right?
Dialogue: 0,0:03:17.76,0:03:21.01,Default,,0000,0000,0000,,Because multiplication is\Nassociative, and order
Dialogue: 0,0:03:21.01,0:03:22.92,Default,,0000,0000,0000,,doesn't matter and all that.
Dialogue: 0,0:03:22.92,0:03:24.54,Default,,0000,0000,0000,,But anyway, what can you\Ndo with it in here?
Dialogue: 0,0:03:24.54,0:03:27.29,Default,,0000,0000,0000,,Well, we can take the integral.
Dialogue: 0,0:03:27.29,0:03:32.52,Default,,0000,0000,0000,,All integrals help us do is\Nhelp us take infinite sums of
Dialogue: 0,0:03:32.52,0:03:36.08,Default,,0000,0000,0000,,infinitely small distances,\Nlike a dz or a dx or
Dialogue: 0,0:03:36.08,0:03:38.24,Default,,0000,0000,0000,,a dy, et cetera.
Dialogue: 0,0:03:38.24,0:03:41.62,Default,,0000,0000,0000,,So, what we could do is we\Ncould take this cube and
Dialogue: 0,0:03:41.62,0:03:44.11,Default,,0000,0000,0000,,first, add it up in, let's\Nsay, the z direction.
Dialogue: 0,0:03:44.11,0:03:48.33,Default,,0000,0000,0000,,So we could take that cube and\Nthen add it along the up and
Dialogue: 0,0:03:48.33,0:03:51.23,Default,,0000,0000,0000,,down axis-- the z-axis--\Nso that we get the
Dialogue: 0,0:03:51.23,0:03:52.41,Default,,0000,0000,0000,,volume of a column.
Dialogue: 0,0:03:52.41,0:03:54.55,Default,,0000,0000,0000,,So what would that look like?
Dialogue: 0,0:03:54.55,0:03:56.93,Default,,0000,0000,0000,,Well, since we're going up and\Ndown, we're adding-- we're
Dialogue: 0,0:03:56.93,0:04:00.67,Default,,0000,0000,0000,,taking the sum in\Nthe z direction.
Dialogue: 0,0:04:00.67,0:04:02.61,Default,,0000,0000,0000,,We'd have an integral.
Dialogue: 0,0:04:02.61,0:04:04.66,Default,,0000,0000,0000,,And then what's the\Nlowest z value?
Dialogue: 0,0:04:04.66,0:04:08.31,Default,,0000,0000,0000,,Well, it's z is equal to 0.
Dialogue: 0,0:04:08.31,0:04:09.28,Default,,0000,0000,0000,,And what's the upper bound?
Dialogue: 0,0:04:09.28,0:04:12.07,Default,,0000,0000,0000,,Like if you were to just take--\Nkeep adding these cubes, and
Dialogue: 0,0:04:12.07,0:04:14.19,Default,,0000,0000,0000,,keep going up, you'd run\Ninto the upper bound.
Dialogue: 0,0:04:14.19,0:04:14.77,Default,,0000,0000,0000,,And what's the upper bound?
Dialogue: 0,0:04:14.77,0:04:16.10,Default,,0000,0000,0000,,It's z is equal to 2.
Dialogue: 0,0:04:16.10,0:04:20.58,Default,,0000,0000,0000,,
Dialogue: 0,0:04:20.58,0:04:25.01,Default,,0000,0000,0000,,And of course, you would\Ntake the sum of these dv's.
Dialogue: 0,0:04:25.01,0:04:26.13,Default,,0000,0000,0000,,And I'll write dz first.
Dialogue: 0,0:04:26.13,0:04:28.17,Default,,0000,0000,0000,,Just so it reminds us\Nthat we're going to
Dialogue: 0,0:04:28.17,0:04:30.43,Default,,0000,0000,0000,,take the integral with\Nrespect to z first.
Dialogue: 0,0:04:30.43,0:04:32.01,Default,,0000,0000,0000,,And let's say we'll do y next.
Dialogue: 0,0:04:32.01,0:04:34.20,Default,,0000,0000,0000,,And then we'll do x.
Dialogue: 0,0:04:34.20,0:04:37.43,Default,,0000,0000,0000,,So this integral, this value,\Nas I've written it, will
Dialogue: 0,0:04:37.43,0:04:42.02,Default,,0000,0000,0000,,figure out the volume of a\Ncolumn given any x and y.
Dialogue: 0,0:04:42.02,0:04:45.24,Default,,0000,0000,0000,,It'll be a function of x and y,\Nbut since we're dealing with
Dialogue: 0,0:04:45.24,0:04:47.13,Default,,0000,0000,0000,,all constants here, it's\Nactually going to be
Dialogue: 0,0:04:47.13,0:04:48.60,Default,,0000,0000,0000,,a constant value.
Dialogue: 0,0:04:48.60,0:04:52.16,Default,,0000,0000,0000,,It'll be the constant value\Nof the volume of one
Dialogue: 0,0:04:52.16,0:04:53.89,Default,,0000,0000,0000,,of these columns.
Dialogue: 0,0:04:53.89,0:04:56.58,Default,,0000,0000,0000,,So essentially, it'll\Nbe 2 times dy dx.
Dialogue: 0,0:04:56.58,0:04:59.33,Default,,0000,0000,0000,,Because the height of one\Nof these columns is 2,
Dialogue: 0,0:04:59.33,0:05:03.71,Default,,0000,0000,0000,,and then its with and\Nits depth is dy and dx.
Dialogue: 0,0:05:03.71,0:05:06.57,Default,,0000,0000,0000,,So then if we want to figure\Nout the entire volume-- what
Dialogue: 0,0:05:06.57,0:05:09.27,Default,,0000,0000,0000,,we did just now is we figured\Nout the height of a column.
Dialogue: 0,0:05:09.27,0:05:11.30,Default,,0000,0000,0000,,So then we could take those\Ncolumns and sum them
Dialogue: 0,0:05:11.30,0:05:13.73,Default,,0000,0000,0000,,in the y direction.
Dialogue: 0,0:05:13.73,0:05:15.71,Default,,0000,0000,0000,,So if we're summing in the y\Ndirection, we could just take
Dialogue: 0,0:05:15.71,0:05:20.34,Default,,0000,0000,0000,,another integral of this\Nsum in the y direction.
Dialogue: 0,0:05:20.34,0:05:25.65,Default,,0000,0000,0000,,And y goes from 0 to what?\Ny goes from 0 to 4.
Dialogue: 0,0:05:25.65,0:05:27.18,Default,,0000,0000,0000,,I wrote this integral a\Nlittle bit too far to the
Dialogue: 0,0:05:27.18,0:05:28.30,Default,,0000,0000,0000,,left, it looks strange.
Dialogue: 0,0:05:28.30,0:05:31.00,Default,,0000,0000,0000,,But I think you get the idea.
Dialogue: 0,0:05:31.00,0:05:33.39,Default,,0000,0000,0000,,y is equal to 0, to\Ny is equal to 4.
Dialogue: 0,0:05:33.39,0:05:37.42,Default,,0000,0000,0000,,And then that'll give us the\Nvolume of a sheet that is
Dialogue: 0,0:05:37.42,0:05:40.29,Default,,0000,0000,0000,,parallel to the zy plane.
Dialogue: 0,0:05:40.29,0:05:44.25,Default,,0000,0000,0000,,And then all we have left to do\Nis add up a bunch of those
Dialogue: 0,0:05:44.25,0:05:46.57,Default,,0000,0000,0000,,sheets in the x direction, and\Nwe'll have the volume
Dialogue: 0,0:05:46.57,0:05:48.21,Default,,0000,0000,0000,,of our entire figure.
Dialogue: 0,0:05:48.21,0:05:50.19,Default,,0000,0000,0000,,So to add up those sheets,\Nwe would have to sum
Dialogue: 0,0:05:50.19,0:05:51.75,Default,,0000,0000,0000,,in the x direction.
Dialogue: 0,0:05:51.75,0:05:57.06,Default,,0000,0000,0000,,And we'd go from x is equal\Nto 0, to x is equal to 3.
Dialogue: 0,0:05:57.06,0:05:58.66,Default,,0000,0000,0000,,And to evaluate this\Nis actually fairly
Dialogue: 0,0:05:58.66,0:05:59.69,Default,,0000,0000,0000,,straightforward.
Dialogue: 0,0:05:59.69,0:06:03.02,Default,,0000,0000,0000,,So, first we're taking the\Nintegral with respect to z.
Dialogue: 0,0:06:03.02,0:06:05.09,Default,,0000,0000,0000,,Well, we don't have anything\Nwritten under here, but we
Dialogue: 0,0:06:05.09,0:06:06.74,Default,,0000,0000,0000,,can just assume that\Nthere's a 1, right?
Dialogue: 0,0:06:06.74,0:06:10.16,Default,,0000,0000,0000,,Because dz times dy times\Ndx is the same thing as
Dialogue: 0,0:06:10.16,0:06:12.94,Default,,0000,0000,0000,,1 times dz times dy dx.
Dialogue: 0,0:06:12.94,0:06:15.50,Default,,0000,0000,0000,,So what's the value\Nof this integral?
Dialogue: 0,0:06:15.50,0:06:18.76,Default,,0000,0000,0000,,Well, the antiderivative\Nof 1 with respect to
Dialogue: 0,0:06:18.76,0:06:20.65,Default,,0000,0000,0000,,z is just z, right?
Dialogue: 0,0:06:20.65,0:06:22.70,Default,,0000,0000,0000,,Because the derivative\Nof z is 1.
Dialogue: 0,0:06:22.70,0:06:27.64,Default,,0000,0000,0000,,And you evaluate\Nthat from 2 to 0.
Dialogue: 0,0:06:27.64,0:06:30.21,Default,,0000,0000,0000,,So then you're left with--\Nso it's 2 minus 0.
Dialogue: 0,0:06:30.21,0:06:31.58,Default,,0000,0000,0000,,So you're just left with 2.
Dialogue: 0,0:06:31.58,0:06:34.39,Default,,0000,0000,0000,,So you're left with 2, and you\Ntake the integral of that from
Dialogue: 0,0:06:34.39,0:06:38.08,Default,,0000,0000,0000,,y is equal to 0, to y is equal\Nto 4 dy, and then
Dialogue: 0,0:06:38.08,0:06:40.06,Default,,0000,0000,0000,,you have the x.
Dialogue: 0,0:06:40.06,0:06:45.28,Default,,0000,0000,0000,,From x is equal to 0,\Nto x is equal to 3 dx.
Dialogue: 0,0:06:45.28,0:06:48.44,Default,,0000,0000,0000,,And notice, when we just took\Nthe integral with respect to
Dialogue: 0,0:06:48.44,0:06:50.21,Default,,0000,0000,0000,,z, we ended up with\Na double integral.
Dialogue: 0,0:06:50.21,0:06:52.83,Default,,0000,0000,0000,,And this double integral is the\Nexact integral we would have
Dialogue: 0,0:06:52.83,0:06:56.44,Default,,0000,0000,0000,,done in the previous videos on\Nthe double integral, where you
Dialogue: 0,0:06:56.44,0:06:59.51,Default,,0000,0000,0000,,would have just said, well,\Nz is a function of x and y.
Dialogue: 0,0:06:59.51,0:07:01.88,Default,,0000,0000,0000,,So you could have written, you\Nknow, z, is a function of x
Dialogue: 0,0:07:01.88,0:07:04.23,Default,,0000,0000,0000,,and y, is always equal to 2.
Dialogue: 0,0:07:04.23,0:07:05.18,Default,,0000,0000,0000,,It's a constant function.
Dialogue: 0,0:07:05.18,0:07:06.98,Default,,0000,0000,0000,,It's independent of x and y.
Dialogue: 0,0:07:06.98,0:07:09.21,Default,,0000,0000,0000,,But if you had defined z in\Nthis way, and you wanted to
Dialogue: 0,0:07:09.21,0:07:11.98,Default,,0000,0000,0000,,figure out the volume under\Nthis surface, where the surface
Dialogue: 0,0:07:11.98,0:07:15.37,Default,,0000,0000,0000,,is z is equal to 2-- you\Nknow, this is a surface, is z
Dialogue: 0,0:07:15.37,0:07:17.58,Default,,0000,0000,0000,,is equal to 2-- we would\Nhave ended up with this.
Dialogue: 0,0:07:17.58,0:07:19.13,Default,,0000,0000,0000,,So you see that what we're\Ndoing with the triple
Dialogue: 0,0:07:19.13,0:07:21.03,Default,,0000,0000,0000,,integral, it's really,\Nreally nothing different.
Dialogue: 0,0:07:21.03,0:07:22.06,Default,,0000,0000,0000,,And you might be wondering,\Nwell, why are we
Dialogue: 0,0:07:22.06,0:07:22.84,Default,,0000,0000,0000,,doing it at all?
Dialogue: 0,0:07:22.84,0:07:25.73,Default,,0000,0000,0000,,And I'll show you\Nthat in a second.
Dialogue: 0,0:07:25.73,0:07:28.32,Default,,0000,0000,0000,,But anyway, to evaluate\Nthis, you could take the
Dialogue: 0,0:07:28.32,0:07:32.07,Default,,0000,0000,0000,,antiderivative of this with\Nrespect to y, you get 2y-- let
Dialogue: 0,0:07:32.07,0:07:33.76,Default,,0000,0000,0000,,me scroll down a little bit.
Dialogue: 0,0:07:33.76,0:07:38.53,Default,,0000,0000,0000,,You get 2y evaluating\Nthat at 4 and 0.
Dialogue: 0,0:07:38.53,0:07:41.15,Default,,0000,0000,0000,,And then, so you get 2 times 4.
Dialogue: 0,0:07:41.15,0:07:42.54,Default,,0000,0000,0000,,So it's 8 minus 0.
Dialogue: 0,0:07:42.54,0:07:46.07,Default,,0000,0000,0000,,And then you integrate\Nthat from, with respect
Dialogue: 0,0:07:46.07,0:07:48.34,Default,,0000,0000,0000,,to x from 0 to 3.
Dialogue: 0,0:07:48.34,0:07:52.43,Default,,0000,0000,0000,,So that's 8x from 0 to 3.
Dialogue: 0,0:07:52.43,0:07:55.43,Default,,0000,0000,0000,,So that'll be equal to\N24 four units cubed.
Dialogue: 0,0:07:55.43,0:07:59.78,Default,,0000,0000,0000,,So I know the obvious question\Nis, what is this good for?
Dialogue: 0,0:07:59.78,0:08:05.42,Default,,0000,0000,0000,,Well, when you have a kind\Nof a constant value within
Dialogue: 0,0:08:05.42,0:08:06.40,Default,,0000,0000,0000,,the volume, you're right.
Dialogue: 0,0:08:06.40,0:08:08.23,Default,,0000,0000,0000,,You could have just done\Na double integral.
Dialogue: 0,0:08:08.23,0:08:11.61,Default,,0000,0000,0000,,But what if I were to tell you,\Nour goal is not to figure out
Dialogue: 0,0:08:11.61,0:08:13.67,Default,,0000,0000,0000,,the volume of this figure.
Dialogue: 0,0:08:13.67,0:08:16.55,Default,,0000,0000,0000,,Our goal is to figure out\Nthe mass of this figure.
Dialogue: 0,0:08:16.55,0:08:21.66,Default,,0000,0000,0000,,And even more, this volume--\Nthis area of space or
Dialogue: 0,0:08:21.66,0:08:23.67,Default,,0000,0000,0000,,whatever-- its mass\Nis not uniform.
Dialogue: 0,0:08:23.67,0:08:28.19,Default,,0000,0000,0000,,If its mass was uniform, you\Ncould just multiply its uniform
Dialogue: 0,0:08:28.19,0:08:31.24,Default,,0000,0000,0000,,density times its volume,\Nand you'd get its mass.
Dialogue: 0,0:08:31.24,0:08:33.04,Default,,0000,0000,0000,,But let's say the\Ndensity changes.
Dialogue: 0,0:08:33.04,0:08:36.34,Default,,0000,0000,0000,,It could be a volume of some\Ngas or it could be even some
Dialogue: 0,0:08:36.34,0:08:39.07,Default,,0000,0000,0000,,material with different\Ncompounds in it.
Dialogue: 0,0:08:39.07,0:08:42.37,Default,,0000,0000,0000,,So let's say that its density\Nis a variable function
Dialogue: 0,0:08:42.37,0:08:43.24,Default,,0000,0000,0000,,of x, y, and z.
Dialogue: 0,0:08:43.24,0:08:47.65,Default,,0000,0000,0000,,So let's say that the density--\Nthis row, this thing that looks
Dialogue: 0,0:08:47.65,0:08:50.72,Default,,0000,0000,0000,,like a p is what you normally\Nuse in physics for density-- so
Dialogue: 0,0:08:50.72,0:08:54.39,Default,,0000,0000,0000,,its density is a function\Nof x, y, and z.
Dialogue: 0,0:08:54.39,0:08:55.71,Default,,0000,0000,0000,,Let's-- just to make it\Nsimple-- let's make
Dialogue: 0,0:08:55.71,0:08:59.84,Default,,0000,0000,0000,,it x times y times z.
Dialogue: 0,0:08:59.84,0:09:06.02,Default,,0000,0000,0000,,If we wanted to figure out the\Nmass of any small volume, it
Dialogue: 0,0:09:06.02,0:09:08.44,Default,,0000,0000,0000,,would be that volume times\Nthe density, right?
Dialogue: 0,0:09:08.44,0:09:12.19,Default,,0000,0000,0000,,Because density-- the units of\Ndensity are like kilograms
Dialogue: 0,0:09:12.19,0:09:13.59,Default,,0000,0000,0000,,per meter cubed.
Dialogue: 0,0:09:13.59,0:09:16.40,Default,,0000,0000,0000,,So if you multiply it times\Nmeter cubed, you get kilograms.
Dialogue: 0,0:09:16.40,0:09:20.26,Default,,0000,0000,0000,,So we could say that the mass--\Nwell, I'll make up notation, d
Dialogue: 0,0:09:20.26,0:09:23.73,Default,,0000,0000,0000,,mass-- this isn't a function.
Dialogue: 0,0:09:23.73,0:09:25.23,Default,,0000,0000,0000,,Well, I don't want to write it\Nin parentheses, because it
Dialogue: 0,0:09:25.23,0:09:26.23,Default,,0000,0000,0000,,makes it look like a function.
Dialogue: 0,0:09:26.23,0:09:30.49,Default,,0000,0000,0000,,So, a very differential mass,\Nor a very small mass, is going
Dialogue: 0,0:09:30.49,0:09:35.86,Default,,0000,0000,0000,,to equal the density at that\Npoint, which would be xyz,
Dialogue: 0,0:09:35.86,0:09:39.81,Default,,0000,0000,0000,,times the volume of that\Nof that small mass.
Dialogue: 0,0:09:39.81,0:09:42.78,Default,,0000,0000,0000,,And that volume of that small\Nmass we could write as dv.
Dialogue: 0,0:09:42.78,0:09:48.79,Default,,0000,0000,0000,,And we know that dv is the\Nsame thing as the width times
Dialogue: 0,0:09:48.79,0:09:49.67,Default,,0000,0000,0000,,the height times the depth.
Dialogue: 0,0:09:49.67,0:09:52.35,Default,,0000,0000,0000,,dv doesn't always have to\Nbe dx times dy times dz.
Dialogue: 0,0:09:52.35,0:09:54.00,Default,,0000,0000,0000,,If we're doing other\Ncoordinates, if we're doing
Dialogue: 0,0:09:54.00,0:09:57.67,Default,,0000,0000,0000,,polar coordinates, it could be\Nsomething slightly different.
Dialogue: 0,0:09:57.67,0:09:59.16,Default,,0000,0000,0000,,And we'll do that eventually.
Dialogue: 0,0:09:59.16,0:10:01.28,Default,,0000,0000,0000,,But if we wanted to figure out\Nthe mass, since we're using
Dialogue: 0,0:10:01.28,0:10:03.55,Default,,0000,0000,0000,,rectangular coordinates, it\Nwould be the density function
Dialogue: 0,0:10:03.55,0:10:07.03,Default,,0000,0000,0000,,at that point times our\Ndifferential volume.
Dialogue: 0,0:10:07.03,0:10:11.33,Default,,0000,0000,0000,,So times dx dy dz.
Dialogue: 0,0:10:11.33,0:10:13.87,Default,,0000,0000,0000,,And of course, we can\Nchange the order here.
Dialogue: 0,0:10:13.87,0:10:16.39,Default,,0000,0000,0000,,So when you want to figure out\Nthe volume-- when you want to
Dialogue: 0,0:10:16.39,0:10:19.00,Default,,0000,0000,0000,,figure out the mass-- which I\Nwill do in the next video, we
Dialogue: 0,0:10:19.00,0:10:21.29,Default,,0000,0000,0000,,essentially will have to\Nintegrate this function.
Dialogue: 0,0:10:21.29,0:10:27.40,Default,,0000,0000,0000,,As opposed to just\N1 over z, y and x.
Dialogue: 0,0:10:27.40,0:10:28.69,Default,,0000,0000,0000,,And I'm going to do that\Nin the next video.
Dialogue: 0,0:10:28.69,0:10:32.05,Default,,0000,0000,0000,,And you'll see that it's really\Njust a lot of basic taking
Dialogue: 0,0:10:32.05,0:10:34.70,Default,,0000,0000,0000,,antiderivatives and avoiding\Ncareless mistakes.
Dialogue: 0,0:10:34.70,0:10:37.28,Default,,0000,0000,0000,,I will see you in\Nthe next video.
Dialogue: 0,0:10:37.28,0:10:37.90,Default,,0000,0000,0000,,