[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.68,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.68,0:00:02.98,Default,,0000,0000,0000,,You hopefully have a little\Nintuition now on what a double
Dialogue: 0,0:00:02.98,0:00:06.92,Default,,0000,0000,0000,,integral is or how we go about\Nfiguring out the volume
Dialogue: 0,0:00:06.92,0:00:07.49,Default,,0000,0000,0000,,under a surface.
Dialogue: 0,0:00:07.49,0:00:09.91,Default,,0000,0000,0000,,So let's actually compute it\Nand I think it'll all become
Dialogue: 0,0:00:09.91,0:00:10.91,Default,,0000,0000,0000,,a lot more concrete.
Dialogue: 0,0:00:10.91,0:00:14.22,Default,,0000,0000,0000,,So let's say I have the\Nsurface, z, and it's a
Dialogue: 0,0:00:14.22,0:00:15.53,Default,,0000,0000,0000,,function of x and y.
Dialogue: 0,0:00:15.53,0:00:20.67,Default,,0000,0000,0000,,And it equals xy squared.
Dialogue: 0,0:00:20.67,0:00:22.85,Default,,0000,0000,0000,,It's a surface in\Nthree-dimensional space.
Dialogue: 0,0:00:22.85,0:00:26.02,Default,,0000,0000,0000,,And I want to know the\Nvolume between this
Dialogue: 0,0:00:26.02,0:00:28.66,Default,,0000,0000,0000,,surface and the xy-plane.
Dialogue: 0,0:00:28.66,0:00:33.32,Default,,0000,0000,0000,,And the domain in the xy-plane\Nthat I care about is x is
Dialogue: 0,0:00:33.32,0:00:38.38,Default,,0000,0000,0000,,greater than or equal to 0,\Nand less than or equal to 2.
Dialogue: 0,0:00:38.38,0:00:42.45,Default,,0000,0000,0000,,And y is greater than or\Nequal to 0, and less
Dialogue: 0,0:00:42.45,0:00:43.74,Default,,0000,0000,0000,,than or equal to 1.
Dialogue: 0,0:00:43.74,0:00:45.37,Default,,0000,0000,0000,,Let's see what that looks\Nlike just so we have a
Dialogue: 0,0:00:45.37,0:00:47.96,Default,,0000,0000,0000,,good visualization of it.
Dialogue: 0,0:00:47.96,0:00:50.26,Default,,0000,0000,0000,,So I graphed it here and\Nwe can rotate it around.
Dialogue: 0,0:00:50.26,0:00:52.75,Default,,0000,0000,0000,,This is z equals xy squared.
Dialogue: 0,0:00:52.75,0:00:56.24,Default,,0000,0000,0000,,This is the bounding box,\Nright? x goes from 0 to
Dialogue: 0,0:00:56.24,0:00:58.30,Default,,0000,0000,0000,,2; y goes from 0 to 1.
Dialogue: 0,0:00:58.30,0:01:00.72,Default,,0000,0000,0000,,We literally want this-- you\Ncould almost view it the
Dialogue: 0,0:01:00.72,0:01:02.71,Default,,0000,0000,0000,,volume-- well, not almost.
Dialogue: 0,0:01:02.71,0:01:05.59,Default,,0000,0000,0000,,Exactly view it as the\Nvolume under this surface.
Dialogue: 0,0:01:05.59,0:01:08.53,Default,,0000,0000,0000,,Between this surface, the top\Nsurface, and the xy-plane.
Dialogue: 0,0:01:08.53,0:01:11.58,Default,,0000,0000,0000,,And I'll rotate it around so\Nyou can get a little bit better
Dialogue: 0,0:01:11.58,0:01:14.21,Default,,0000,0000,0000,,sense of the actual volume.
Dialogue: 0,0:01:14.21,0:01:16.25,Default,,0000,0000,0000,,Let me rotate.
Dialogue: 0,0:01:16.25,0:01:19.33,Default,,0000,0000,0000,,Now I should use the\Nmouse for this.
Dialogue: 0,0:01:19.33,0:01:21.38,Default,,0000,0000,0000,,So it's this space,\Nunderneath here.
Dialogue: 0,0:01:21.38,0:01:23.98,Default,,0000,0000,0000,,It's like a makeshift\Nshelter or something.
Dialogue: 0,0:01:23.98,0:01:27.06,Default,,0000,0000,0000,,I could rotate it a little bit.
Dialogue: 0,0:01:27.06,0:01:29.34,Default,,0000,0000,0000,,Whatever's under this,\Nbetween the two surfaces--
Dialogue: 0,0:01:29.34,0:01:30.92,Default,,0000,0000,0000,,that's the volume.
Dialogue: 0,0:01:30.92,0:01:32.55,Default,,0000,0000,0000,,Whoops, I've flipped it.
Dialogue: 0,0:01:32.55,0:01:33.50,Default,,0000,0000,0000,,There you go.
Dialogue: 0,0:01:33.50,0:01:35.69,Default,,0000,0000,0000,,So that's the volume\Nthat we care about.
Dialogue: 0,0:01:35.69,0:01:38.49,Default,,0000,0000,0000,,Let's figure out how to do and\Nwe'll try to gather a little
Dialogue: 0,0:01:38.49,0:01:41.48,Default,,0000,0000,0000,,bit of the intuition\Nas we go along.
Dialogue: 0,0:01:41.48,0:01:44.85,Default,,0000,0000,0000,,So I'm going to draw a not as\Nimpressive version of that
Dialogue: 0,0:01:44.85,0:01:49.03,Default,,0000,0000,0000,,graph, but I think it'll\Ndo the job for now.
Dialogue: 0,0:01:49.03,0:01:50.18,Default,,0000,0000,0000,,Let me draw my axes.
Dialogue: 0,0:01:50.18,0:01:52.71,Default,,0000,0000,0000,,
Dialogue: 0,0:01:52.71,0:02:01.03,Default,,0000,0000,0000,,That's my x-axis, that's my\Ny-axis, and that's my z-axis.
Dialogue: 0,0:02:01.03,0:02:04.55,Default,,0000,0000,0000,,
Dialogue: 0,0:02:04.55,0:02:08.81,Default,,0000,0000,0000,,x, y, z.
Dialogue: 0,0:02:08.81,0:02:10.87,Default,,0000,0000,0000,,x is going from 0 to 2.
Dialogue: 0,0:02:10.87,0:02:12.30,Default,,0000,0000,0000,,Let's say that's 2.
Dialogue: 0,0:02:12.30,0:02:16.16,Default,,0000,0000,0000,,y is going from 0 to 1.
Dialogue: 0,0:02:16.16,0:02:20.80,Default,,0000,0000,0000,,So we're taking the volume\Nabove this rectangle
Dialogue: 0,0:02:20.80,0:02:23.57,Default,,0000,0000,0000,,in the xy-plane.
Dialogue: 0,0:02:23.57,0:02:25.74,Default,,0000,0000,0000,,And then the surface, I'm going\Nto try my best to draw it.
Dialogue: 0,0:02:25.74,0:02:27.66,Default,,0000,0000,0000,,I'll draw it in a\Ndifferent color.
Dialogue: 0,0:02:27.66,0:02:30.68,Default,,0000,0000,0000,,I'm looking at the picture.
Dialogue: 0,0:02:30.68,0:02:32.60,Default,,0000,0000,0000,,At this end it looks\Nsomething like this.
Dialogue: 0,0:02:32.60,0:02:36.30,Default,,0000,0000,0000,,
Dialogue: 0,0:02:36.30,0:02:37.74,Default,,0000,0000,0000,,And then it has a\Nstraight line.
Dialogue: 0,0:02:37.74,0:02:43.58,Default,,0000,0000,0000,,Let's see if I can draw this\Nsurface going down like that.
Dialogue: 0,0:02:43.58,0:02:47.18,Default,,0000,0000,0000,,And then if I was really\Ngood I could shade it.
Dialogue: 0,0:02:47.18,0:02:50.70,Default,,0000,0000,0000,,It looks something like this.
Dialogue: 0,0:02:50.70,0:02:55.74,Default,,0000,0000,0000,,If I were to shade it,\Nthe surface looks
Dialogue: 0,0:02:55.74,0:02:57.02,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:02:57.02,0:02:59.78,Default,,0000,0000,0000,,And this right here\Nis above this.
Dialogue: 0,0:02:59.78,0:03:04.38,Default,,0000,0000,0000,,This is the bottom left corner,\Nyou can almost view it.
Dialogue: 0,0:03:04.38,0:03:08.70,Default,,0000,0000,0000,,So let me just say the yellow\Nis the top of the surface.
Dialogue: 0,0:03:08.70,0:03:09.83,Default,,0000,0000,0000,,That's the top of the surface.
Dialogue: 0,0:03:09.83,0:03:11.83,Default,,0000,0000,0000,,And then this is\Nunder the surface.
Dialogue: 0,0:03:11.83,0:03:15.26,Default,,0000,0000,0000,,So we care about this\Nvolume underneath here.
Dialogue: 0,0:03:15.26,0:03:17.84,Default,,0000,0000,0000,,Let me show you the\Nactual surface.
Dialogue: 0,0:03:17.84,0:03:20.28,Default,,0000,0000,0000,,So this volume underneath here.
Dialogue: 0,0:03:20.28,0:03:21.06,Default,,0000,0000,0000,,I think you get the idea.
Dialogue: 0,0:03:21.06,0:03:22.56,Default,,0000,0000,0000,,So how do we do that?
Dialogue: 0,0:03:22.56,0:03:26.59,Default,,0000,0000,0000,,Well, in the last example we\Nsaid, well, let's pick an
Dialogue: 0,0:03:26.59,0:03:29.92,Default,,0000,0000,0000,,arbitrary y and for that\Ny, let's figure out the
Dialogue: 0,0:03:29.92,0:03:31.25,Default,,0000,0000,0000,,area under the curve.
Dialogue: 0,0:03:31.25,0:03:36.28,Default,,0000,0000,0000,,So if we fix some y-- when you\Nactually do the problem, you
Dialogue: 0,0:03:36.28,0:03:39.55,Default,,0000,0000,0000,,don't have to think about it in\Nthis much detail, but I want
Dialogue: 0,0:03:39.55,0:03:40.41,Default,,0000,0000,0000,,to give you the intuition.
Dialogue: 0,0:03:40.41,0:03:43.81,Default,,0000,0000,0000,,So if we pick just an\Narbitrary y here.
Dialogue: 0,0:03:43.81,0:03:48.25,Default,,0000,0000,0000,,So on that y, we could think of\Nit-- if we have a fixed y, then
Dialogue: 0,0:03:48.25,0:03:51.48,Default,,0000,0000,0000,,the function of x and y you can\Nalmost view it as a function
Dialogue: 0,0:03:51.48,0:03:56.62,Default,,0000,0000,0000,,of just x for that given y.
Dialogue: 0,0:03:56.62,0:04:02.61,Default,,0000,0000,0000,,And so, we're kind of figuring\Nout the value of this, of
Dialogue: 0,0:04:02.61,0:04:04.47,Default,,0000,0000,0000,,the area under this curve.
Dialogue: 0,0:04:04.47,0:04:08.43,Default,,0000,0000,0000,,
Dialogue: 0,0:04:08.43,0:04:11.82,Default,,0000,0000,0000,,You should view this as kind of\Nan up down curve for a given y.
Dialogue: 0,0:04:11.82,0:04:15.87,Default,,0000,0000,0000,,So if we know a y we can figure\Nout then-- for example, if y
Dialogue: 0,0:04:15.87,0:04:20.20,Default,,0000,0000,0000,,was 5, this function would\Nbecome z equals 25x.
Dialogue: 0,0:04:20.20,0:04:22.57,Default,,0000,0000,0000,,And then that's easy to\Nfigure out the value
Dialogue: 0,0:04:22.57,0:04:23.35,Default,,0000,0000,0000,,of the curve under.
Dialogue: 0,0:04:23.35,0:04:26.07,Default,,0000,0000,0000,,So we'll make the value under\Nthe curve as a function of y.
Dialogue: 0,0:04:26.07,0:04:27.50,Default,,0000,0000,0000,,We'll pretend like\Nit's just a constant.
Dialogue: 0,0:04:27.50,0:04:28.77,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:04:28.77,0:04:33.68,Default,,0000,0000,0000,,So if we have a dx\Nthat's our change in x.
Dialogue: 0,0:04:33.68,0:04:36.71,Default,,0000,0000,0000,,And then our height of each of\Nour rectangles is going to be a
Dialogue: 0,0:04:36.71,0:04:40.01,Default,,0000,0000,0000,,function-- it's going to be z.
Dialogue: 0,0:04:40.01,0:04:42.66,Default,,0000,0000,0000,,The height is z, which is\Na function of x and y.
Dialogue: 0,0:04:42.66,0:04:45.19,Default,,0000,0000,0000,,So we can take the integral.
Dialogue: 0,0:04:45.19,0:04:50.02,Default,,0000,0000,0000,,So the area of each of these is\Ngoing to be our function, xy
Dialogue: 0,0:04:50.02,0:04:54.76,Default,,0000,0000,0000,,squared-- I'll do it here\Nbecause I'll run out of space.
Dialogue: 0,0:04:54.76,0:04:59.02,Default,,0000,0000,0000,,xy squared times the\Nwidth, which is dx.
Dialogue: 0,0:04:59.02,0:05:05.71,Default,,0000,0000,0000,,And if we want the area of this\Nslice for a given y, we just
Dialogue: 0,0:05:05.71,0:05:08.03,Default,,0000,0000,0000,,integrate along the x-axis.
Dialogue: 0,0:05:08.03,0:05:10.10,Default,,0000,0000,0000,,We're going to integrate\Nfrom x is equal to 0
Dialogue: 0,0:05:10.10,0:05:12.23,Default,,0000,0000,0000,,to x is equal to 2.
Dialogue: 0,0:05:12.23,0:05:15.21,Default,,0000,0000,0000,,From x is equal to 0 to 2.
Dialogue: 0,0:05:15.21,0:05:16.79,Default,,0000,0000,0000,,Fair enough.
Dialogue: 0,0:05:16.79,0:05:21.05,Default,,0000,0000,0000,,Now, but we just don't want to\Nfigure out the area under the
Dialogue: 0,0:05:21.05,0:05:23.60,Default,,0000,0000,0000,,curve at one slice, for one\Nparticular y, we want to
Dialogue: 0,0:05:23.60,0:05:25.83,Default,,0000,0000,0000,,figure out the entire\Narea of the curve.
Dialogue: 0,0:05:25.83,0:05:27.57,Default,,0000,0000,0000,,So what we do is\Nwe say, OK, fine.
Dialogue: 0,0:05:27.57,0:05:33.37,Default,,0000,0000,0000,,The area under the curve, not\Nthe surface-- under this curve
Dialogue: 0,0:05:33.37,0:05:37.05,Default,,0000,0000,0000,,for a particular y,\Nis this expression.
Dialogue: 0,0:05:37.05,0:05:40.55,Default,,0000,0000,0000,,Well, what if I gave it\Na little bit of depth?
Dialogue: 0,0:05:40.55,0:05:45.54,Default,,0000,0000,0000,,If I multiplied this area times\Ndy then it would give me a
Dialogue: 0,0:05:45.54,0:05:46.85,Default,,0000,0000,0000,,little bit of depth, right?
Dialogue: 0,0:05:46.85,0:05:50.14,Default,,0000,0000,0000,,We'd kind of have a\Nthree-dimensional slice of the
Dialogue: 0,0:05:50.14,0:05:51.24,Default,,0000,0000,0000,,volume that we care about.
Dialogue: 0,0:05:51.24,0:05:52.87,Default,,0000,0000,0000,,I know it's hard to imagine.
Dialogue: 0,0:05:52.87,0:05:54.35,Default,,0000,0000,0000,,Let me bring that here.
Dialogue: 0,0:05:54.35,0:05:58.56,Default,,0000,0000,0000,,So if I had a slice here, we\Njust figured out the area of a
Dialogue: 0,0:05:58.56,0:06:01.40,Default,,0000,0000,0000,,slice and then I'm multiplying\Nit by dy to give it a
Dialogue: 0,0:06:01.40,0:06:04.20,Default,,0000,0000,0000,,little bit of depth.
Dialogue: 0,0:06:04.20,0:06:08.00,Default,,0000,0000,0000,,So you multiply it by dy to\Ngive it a little bit of depth,
Dialogue: 0,0:06:08.00,0:06:11.55,Default,,0000,0000,0000,,and then if we want the entire\Nvolume under the curve we add
Dialogue: 0,0:06:11.55,0:06:14.07,Default,,0000,0000,0000,,all the dy's together, take the\Ninfinite sum of these
Dialogue: 0,0:06:14.07,0:06:17.30,Default,,0000,0000,0000,,infinitely small\Nvolumes really now.
Dialogue: 0,0:06:17.30,0:06:21.45,Default,,0000,0000,0000,,And so we will integrate\Nfrom y is equal to 0
Dialogue: 0,0:06:21.45,0:06:22.57,Default,,0000,0000,0000,,to y is equal to 1.
Dialogue: 0,0:06:22.57,0:06:24.29,Default,,0000,0000,0000,,I know this graph is a little\Nhard to understand, but you
Dialogue: 0,0:06:24.29,0:06:27.18,Default,,0000,0000,0000,,might want to re-watch\Nthe first video.
Dialogue: 0,0:06:27.18,0:06:30.54,Default,,0000,0000,0000,,I had a slightly easier\Nto understand surface.
Dialogue: 0,0:06:30.54,0:06:33.59,Default,,0000,0000,0000,,So now, how do we\Nevaluate this?
Dialogue: 0,0:06:33.59,0:06:36.51,Default,,0000,0000,0000,,Well, like we said,\Nyou evaluate from the
Dialogue: 0,0:06:36.51,0:06:37.50,Default,,0000,0000,0000,,inside and go outward.
Dialogue: 0,0:06:37.50,0:06:40.48,Default,,0000,0000,0000,,
Dialogue: 0,0:06:40.48,0:06:43.51,Default,,0000,0000,0000,,It's taking a partial\Nderivative in reverse.
Dialogue: 0,0:06:43.51,0:06:47.54,Default,,0000,0000,0000,,So we're integrating here with\Nrespect to x, so we can treat
Dialogue: 0,0:06:47.54,0:06:49.42,Default,,0000,0000,0000,,y just like a constant.
Dialogue: 0,0:06:49.42,0:06:51.67,Default,,0000,0000,0000,,Like it's like the number\N5 or something like that.
Dialogue: 0,0:06:51.67,0:06:53.62,Default,,0000,0000,0000,,So it really doesn't\Nchange the integral.
Dialogue: 0,0:06:53.62,0:06:57.06,Default,,0000,0000,0000,,So what's the antiderivative\Nof xy squared?
Dialogue: 0,0:06:57.06,0:07:00.16,Default,,0000,0000,0000,,Well, the antiderivative of\Nxy squared-- I want to make
Dialogue: 0,0:07:00.16,0:07:02.28,Default,,0000,0000,0000,,sure I'm color consistent.
Dialogue: 0,0:07:02.28,0:07:05.72,Default,,0000,0000,0000,,Well, the antiderivative\Nof x is x to the 1/2--
Dialogue: 0,0:07:05.72,0:07:09.08,Default,,0000,0000,0000,,sorry. x squared over 2.
Dialogue: 0,0:07:09.08,0:07:12.18,Default,,0000,0000,0000,,And then y squared is\Njust a constant, right?
Dialogue: 0,0:07:12.18,0:07:14.58,Default,,0000,0000,0000,,And then we don't have to\Nworry about plus c since
Dialogue: 0,0:07:14.58,0:07:15.96,Default,,0000,0000,0000,,this is a definite integral.
Dialogue: 0,0:07:15.96,0:07:18.99,Default,,0000,0000,0000,,And we're going to\Nevaluate that at 2 and 0.
Dialogue: 0,0:07:18.99,0:07:21.19,Default,,0000,0000,0000,,And then we still have\Nthe outside integral
Dialogue: 0,0:07:21.19,0:07:22.65,Default,,0000,0000,0000,,with respect to y.
Dialogue: 0,0:07:22.65,0:07:25.19,Default,,0000,0000,0000,,So once we figure that out\Nwe're going to integrate it
Dialogue: 0,0:07:25.19,0:07:29.80,Default,,0000,0000,0000,,from 0 to 1 with respect to dy.
Dialogue: 0,0:07:29.80,0:07:31.47,Default,,0000,0000,0000,,Now what does this evaluate?
Dialogue: 0,0:07:31.47,0:07:32.91,Default,,0000,0000,0000,,We put a 2 in here.
Dialogue: 0,0:07:32.91,0:07:36.28,Default,,0000,0000,0000,,If you put a 2 in there\Nyou get 2 squared over 2.
Dialogue: 0,0:07:36.28,0:07:39.23,Default,,0000,0000,0000,,
Dialogue: 0,0:07:39.23,0:07:41.74,Default,,0000,0000,0000,,That's just 4 over 2.
Dialogue: 0,0:07:41.74,0:07:43.56,Default,,0000,0000,0000,,So it's 2 y squared.
Dialogue: 0,0:07:43.56,0:07:47.67,Default,,0000,0000,0000,,
Dialogue: 0,0:07:47.67,0:07:51.21,Default,,0000,0000,0000,,Minus 0 squared over\N2 times y squared.
Dialogue: 0,0:07:51.21,0:07:52.08,Default,,0000,0000,0000,,Well, that's just\Ngoing to be 0.
Dialogue: 0,0:07:52.08,0:07:52.95,Default,,0000,0000,0000,,So it's minus 0.
Dialogue: 0,0:07:52.95,0:07:55.22,Default,,0000,0000,0000,,I won't write that down because\Nhopefully that's a little
Dialogue: 0,0:07:55.22,0:07:56.19,Default,,0000,0000,0000,,bit of second nature to you.
Dialogue: 0,0:07:56.19,0:07:58.51,Default,,0000,0000,0000,,We just evaluated this\Nat the 2 endpoints and
Dialogue: 0,0:07:58.51,0:08:00.66,Default,,0000,0000,0000,,I'm short for space.
Dialogue: 0,0:08:00.66,0:08:03.71,Default,,0000,0000,0000,,So this evaluated at 2y\Nsquared and now we evaluate
Dialogue: 0,0:08:03.71,0:08:05.58,Default,,0000,0000,0000,,the outside integral.
Dialogue: 0,0:08:05.58,0:08:08.91,Default,,0000,0000,0000,,0, 1 dy.
Dialogue: 0,0:08:08.91,0:08:10.23,Default,,0000,0000,0000,,And this is an important\Nthing to realize.
Dialogue: 0,0:08:10.23,0:08:13.12,Default,,0000,0000,0000,,When we evaluated this\Ninside integral, remember
Dialogue: 0,0:08:13.12,0:08:13.82,Default,,0000,0000,0000,,what we were doing?
Dialogue: 0,0:08:13.82,0:08:16.95,Default,,0000,0000,0000,,We were trying to figure out\Nfor a given y, what the
Dialogue: 0,0:08:16.95,0:08:19.18,Default,,0000,0000,0000,,area of this surface was.
Dialogue: 0,0:08:19.18,0:08:23.07,Default,,0000,0000,0000,,Well, not this surface, the\Narea under the surface
Dialogue: 0,0:08:23.07,0:08:24.38,Default,,0000,0000,0000,,for a given y.
Dialogue: 0,0:08:24.38,0:08:27.19,Default,,0000,0000,0000,,For a given y that surface\Nkind of turns into a curve.
Dialogue: 0,0:08:27.19,0:08:30.11,Default,,0000,0000,0000,,And we tried to figure out\Nthe area under that curve
Dialogue: 0,0:08:30.11,0:08:33.54,Default,,0000,0000,0000,,in the traditional sense.
Dialogue: 0,0:08:33.54,0:08:36.87,Default,,0000,0000,0000,,This ended up being\Na function of y.
Dialogue: 0,0:08:36.87,0:08:40.50,Default,,0000,0000,0000,,And that makes sense because\Ndepending on which y we pick
Dialogue: 0,0:08:40.50,0:08:44.39,Default,,0000,0000,0000,,we're going to get a\Ndifferent area here.
Dialogue: 0,0:08:44.39,0:08:47.81,Default,,0000,0000,0000,,Obviously, depending on which y\Nwe pick, the area-- kind of a
Dialogue: 0,0:08:47.81,0:08:52.62,Default,,0000,0000,0000,,wall dropped straight down--\Nthat area's going to change.
Dialogue: 0,0:08:52.62,0:08:55.76,Default,,0000,0000,0000,,So we got a function of y when\Nwe evaluated this and now we
Dialogue: 0,0:08:55.76,0:08:58.33,Default,,0000,0000,0000,,just integrate with respect to\Ny and this is just plain old
Dialogue: 0,0:08:58.33,0:09:00.81,Default,,0000,0000,0000,,vanilla definite integration.
Dialogue: 0,0:09:00.81,0:09:03.35,Default,,0000,0000,0000,,What's the antiderivative\Nof 2y squared?
Dialogue: 0,0:09:03.35,0:09:08.14,Default,,0000,0000,0000,,Well, that equals 2 times\Ny to the third over 3,
Dialogue: 0,0:09:08.14,0:09:11.51,Default,,0000,0000,0000,,or 2/3 y to the third.
Dialogue: 0,0:09:11.51,0:09:14.74,Default,,0000,0000,0000,,We're going to evaluate\Nthat at 1 and 0, which
Dialogue: 0,0:09:14.74,0:09:16.10,Default,,0000,0000,0000,,is equal to-- let's see.
Dialogue: 0,0:09:16.10,0:09:17.48,Default,,0000,0000,0000,,1 to the third times 2/3.
Dialogue: 0,0:09:17.48,0:09:18.87,Default,,0000,0000,0000,,That's 2/3.
Dialogue: 0,0:09:18.87,0:09:20.46,Default,,0000,0000,0000,,Minus 0 to the third times 2/3.
Dialogue: 0,0:09:20.46,0:09:21.58,Default,,0000,0000,0000,,Well, that's just 0.
Dialogue: 0,0:09:21.58,0:09:25.27,Default,,0000,0000,0000,,So it equals 2/3.
Dialogue: 0,0:09:25.27,0:09:29.62,Default,,0000,0000,0000,,If our units were meters\Nthese would be 2/3 meters
Dialogue: 0,0:09:29.62,0:09:31.23,Default,,0000,0000,0000,,cubed or cubic meters.
Dialogue: 0,0:09:31.23,0:09:32.28,Default,,0000,0000,0000,,But there you go.
Dialogue: 0,0:09:32.28,0:09:34.89,Default,,0000,0000,0000,,That's how you evaluate\Na double integral.
Dialogue: 0,0:09:34.89,0:09:36.45,Default,,0000,0000,0000,,There really isn't\Na new skill here.
Dialogue: 0,0:09:36.45,0:09:38.65,Default,,0000,0000,0000,,You just have to make sure to\Nkeep track of the variables.
Dialogue: 0,0:09:38.65,0:09:39.76,Default,,0000,0000,0000,,Treat them constant.
Dialogue: 0,0:09:39.76,0:09:41.62,Default,,0000,0000,0000,,They need to be treated\Nconstant, and then treat them
Dialogue: 0,0:09:41.62,0:09:44.71,Default,,0000,0000,0000,,as a variable of integration\Nwhen it's appropriate.
Dialogue: 0,0:09:44.71,0:09:49.09,Default,,0000,0000,0000,,Anyway, I will see you\Nin the next video.
Dialogue: 0,0:09:49.09,0:09:49.90,Default,,0000,0000,0000,,