[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.01,0:00:06.70,Default,,0000,0000,0000,,By now you have had a lot\Nof opportunity to practice
Dialogue: 0,0:00:06.70,0:00:07.97,Default,,0000,0000,0000,,differentiating common\Nfunctions.
Dialogue: 0,0:00:09.06,0:00:12.12,Default,,0000,0000,0000,,And you have learned many\Ntechniques of Differentiation.
Dialogue: 0,0:00:12.98,0:00:16.68,Default,,0000,0000,0000,,So for example, if I were to\Ngive you a function, let's call
Dialogue: 0,0:00:16.68,0:00:18.40,Default,,0000,0000,0000,,this function capital F of X.
Dialogue: 0,0:00:21.15,0:00:23.51,Default,,0000,0000,0000,,If it's a fairly simple\Nfunction, you'll know
Dialogue: 0,0:00:23.51,0:00:24.69,Default,,0000,0000,0000,,how to differentiate it.
Dialogue: 0,0:00:34.76,0:00:39.56,Default,,0000,0000,0000,,And so by differentiating it,\Nyou'll be able to calculate its
Dialogue: 0,0:00:39.56,0:00:43.92,Default,,0000,0000,0000,,derivative, which you'll\Nremember we denote as DF DX. So
Dialogue: 0,0:00:43.92,0:00:47.84,Default,,0000,0000,0000,,that's a process that you're\Nvery familiar with already.
Dialogue: 0,0:00:48.86,0:00:54.91,Default,,0000,0000,0000,,Now in this unit we're going to\Nrefer to DF DX as little F of X.
Dialogue: 0,0:00:57.62,0:00:59.49,Default,,0000,0000,0000,,So little F of X.
Dialogue: 0,0:01:00.21,0:01:04.15,Default,,0000,0000,0000,,Is the derivative of big\NF of X. Let me write
Dialogue: 0,0:01:04.15,0:01:06.30,Default,,0000,0000,0000,,that down little F of X.
Dialogue: 0,0:01:07.83,0:01:09.62,Default,,0000,0000,0000,,Is the derivative.
Dialogue: 0,0:01:18.24,0:01:19.49,Default,,0000,0000,0000,,As big banks.
Dialogue: 0,0:01:20.80,0:01:24.65,Default,,0000,0000,0000,,And as I say, that's a process\Nthat you're very familiar with.
Dialogue: 0,0:01:24.65,0:01:29.15,Default,,0000,0000,0000,,What we want to do now is try to\Nwork this process in reverse.
Dialogue: 0,0:01:29.15,0:01:32.68,Default,,0000,0000,0000,,Work it backwards. In other\Nwords, we want to start with
Dialogue: 0,0:01:32.68,0:01:36.50,Default,,0000,0000,0000,,little F. And try and\Ncome back this route.
Dialogue: 0,0:01:38.81,0:01:42.89,Default,,0000,0000,0000,,And try and find the function or\Nfunctions capital F which when
Dialogue: 0,0:01:42.89,0:01:46.29,Default,,0000,0000,0000,,they are differentiated will\Ngive you little Earth. So we
Dialogue: 0,0:01:46.29,0:01:50.71,Default,,0000,0000,0000,,want to carry out this process\Nin reverse. We can think of this
Dialogue: 0,0:01:50.71,0:01:51.73,Default,,0000,0000,0000,,as anti differentiation.
Dialogue: 0,0:01:59.05,0:02:03.09,Default,,0000,0000,0000,,So we can think of this anti\Ndifferentiation as
Dialogue: 0,0:02:03.09,0:02:04.44,Default,,0000,0000,0000,,differentiation in reverse.
Dialogue: 0,0:02:05.77,0:02:10.58,Default,,0000,0000,0000,,So big F of X we're going to\Ncall the anti derivative of
Dialogue: 0,0:02:10.58,0:02:12.06,Default,,0000,0000,0000,,little F of X.
Dialogue: 0,0:02:26.86,0:02:30.58,Default,,0000,0000,0000,,So by these two results in mind\Nand try not to get confused with
Dialogue: 0,0:02:30.58,0:02:34.31,Default,,0000,0000,0000,,a capital F in the lower case F\Nlittle F is the derivative of
Dialogue: 0,0:02:34.31,0:02:40.01,Default,,0000,0000,0000,,Big F. Because little F\Nis DF DX.
Dialogue: 0,0:02:41.65,0:02:45.42,Default,,0000,0000,0000,,And big F is the anti derivative\Nof little laugh.
Dialogue: 0,0:02:47.20,0:02:50.98,Default,,0000,0000,0000,,Now you may have already looked\Nat a previous video called
Dialogue: 0,0:02:50.98,0:02:52.02,Default,,0000,0000,0000,,integration as summation.
Dialogue: 0,0:02:53.12,0:02:57.56,Default,,0000,0000,0000,,And in that video, the concept\Nof an integral is defined in
Dialogue: 0,0:02:57.56,0:03:02.00,Default,,0000,0000,0000,,terms of the sum of lots of\Nrectangular areas under a curve.
Dialogue: 0,0:03:02.94,0:03:07.11,Default,,0000,0000,0000,,To calculate an integral, you\Nneed to find the limit of a son.
Dialogue: 0,0:03:07.70,0:03:10.72,Default,,0000,0000,0000,,And that's a very cumbersome\Nand impractical process. What
Dialogue: 0,0:03:10.72,0:03:14.76,Default,,0000,0000,0000,,we're going to learn about in\Nthis video is how to find
Dialogue: 0,0:03:14.76,0:03:18.45,Default,,0000,0000,0000,,integrals, not by finding the\Nlimit of a sum, but instead
Dialogue: 0,0:03:18.45,0:03:19.46,Default,,0000,0000,0000,,by using antiderivatives.
Dialogue: 0,0:03:20.52,0:03:22.67,Default,,0000,0000,0000,,Let's start off\Nwith the example.
Dialogue: 0,0:03:24.11,0:03:27.97,Default,,0000,0000,0000,,Suppose we start off with the\Nfunction capital F of X.
Dialogue: 0,0:03:29.66,0:03:34.08,Default,,0000,0000,0000,,Is equal to three\NX squared plus 7X.
Dialogue: 0,0:03:36.31,0:03:37.19,Default,,0000,0000,0000,,Minus 2.
Dialogue: 0,0:03:38.68,0:03:41.36,Default,,0000,0000,0000,,And what I'm going to do is\Nsomething that you're already
Dialogue: 0,0:03:41.36,0:03:43.32,Default,,0000,0000,0000,,very familiar with. We're\Ngoing to differentiate it.
Dialogue: 0,0:03:44.72,0:03:46.35,Default,,0000,0000,0000,,And if we differentiate it.
Dialogue: 0,0:03:47.29,0:03:50.80,Default,,0000,0000,0000,,We get TF DX equals.
Dialogue: 0,0:03:51.65,0:03:55.36,Default,,0000,0000,0000,,Turn by turn the derivative of\Nthree X squared is going to be.
Dialogue: 0,0:03:55.94,0:03:57.47,Default,,0000,0000,0000,,2306 X.
Dialogue: 0,0:04:00.04,0:04:03.23,Default,,0000,0000,0000,,And the derivative of Seven X is\Njust going to be 7.
Dialogue: 0,0:04:05.23,0:04:09.18,Default,,0000,0000,0000,,What about the minus two? Well,\Nyou remember that the derivative
Dialogue: 0,0:04:09.18,0:04:13.49,Default,,0000,0000,0000,,of a constant is 0, so when we\Ndo this differentiation process,
Dialogue: 0,0:04:13.49,0:04:14.92,Default,,0000,0000,0000,,the minus two disappears.
Dialogue: 0,0:04:15.49,0:04:19.31,Default,,0000,0000,0000,,So our derivative DF DX is just\Nsix X +7.
Dialogue: 0,0:04:20.77,0:04:25.45,Default,,0000,0000,0000,,So I'm going to write little F\Nof X is 6 X +7.
Dialogue: 0,0:04:26.98,0:04:31.34,Default,,0000,0000,0000,,And think about what will happen\Nwhen we reverse the process when
Dialogue: 0,0:04:31.34,0:04:32.79,Default,,0000,0000,0000,,we do anti differentiation.
Dialogue: 0,0:04:33.92,0:04:38.42,Default,,0000,0000,0000,,Ask yourself what is an anti\Nderivative of 6X plus 7?
Dialogue: 0,0:04:56.10,0:04:59.56,Default,,0000,0000,0000,,Now we already have an answer to\Nthis question and anti
Dialogue: 0,0:04:59.56,0:05:01.14,Default,,0000,0000,0000,,derivative of 6X plus 7.
Dialogue: 0,0:05:02.15,0:05:05.88,Default,,0000,0000,0000,,Is 3 X squared plus 7X minus\Ntwo, so the answer.
Dialogue: 0,0:05:07.53,0:05:13.76,Default,,0000,0000,0000,,Is F of X is 3 X squared plus\N7X minus two, but I'm afraid
Dialogue: 0,0:05:13.76,0:05:18.32,Default,,0000,0000,0000,,that's not the whole story.\NWe've already seen that when we
Dialogue: 0,0:05:18.32,0:05:22.47,Default,,0000,0000,0000,,differentiate it, three X\Nsquared plus 7X minus two, the
Dialogue: 0,0:05:22.47,0:05:23.72,Default,,0000,0000,0000,,minus two disappeared.
Dialogue: 0,0:05:24.79,0:05:28.41,Default,,0000,0000,0000,,In other words, whatever number\Nhad been in here, whether that
Dialogue: 0,0:05:28.41,0:05:33.02,Default,,0000,0000,0000,,minus two had been minus 8 or\Nplus 10 or 0, whatever it would
Dialogue: 0,0:05:33.02,0:05:35.65,Default,,0000,0000,0000,,still have disappeared. We've\Nlost some information during
Dialogue: 0,0:05:35.65,0:05:39.27,Default,,0000,0000,0000,,this process of Differentiation,\Nand when we want to reverse it
Dialogue: 0,0:05:39.27,0:05:43.54,Default,,0000,0000,0000,,when we want to start with the\Nsix, XX have Seven, and working
Dialogue: 0,0:05:43.54,0:05:47.16,Default,,0000,0000,0000,,backwards, we've really no idea\Nwhat that minus two might have
Dialogue: 0,0:05:47.16,0:05:49.14,Default,,0000,0000,0000,,been. It could have been another
Dialogue: 0,0:05:49.14,0:05:54.38,Default,,0000,0000,0000,,number. So this leads us to the\Nconclusion that once we found an
Dialogue: 0,0:05:54.38,0:05:57.89,Default,,0000,0000,0000,,antiderivative like this one,\Nthree X squared plus, 7X minus
Dialogue: 0,0:05:57.89,0:06:02.53,Default,,0000,0000,0000,,2. Than any constant added\Nonto this will still be an
Dialogue: 0,0:06:02.53,0:06:03.27,Default,,0000,0000,0000,,anti derivative.
Dialogue: 0,0:06:05.79,0:06:07.11,Default,,0000,0000,0000,,If F of X.
Dialogue: 0,0:06:08.46,0:06:10.77,Default,,0000,0000,0000,,Is an anti derivative.
Dialogue: 0,0:06:16.64,0:06:18.18,Default,,0000,0000,0000,,A little F of X.
Dialogue: 0,0:06:19.26,0:06:20.56,Default,,0000,0000,0000,,Then so too.
Dialogue: 0,0:06:24.07,0:06:28.74,Default,,0000,0000,0000,,Is F of X the anti derivative\Nwe've found plus any constant at
Dialogue: 0,0:06:28.74,0:06:33.04,Default,,0000,0000,0000,,all we choose. See is what we\Ncall an arbitrary constant. Any
Dialogue: 0,0:06:33.04,0:06:37.71,Default,,0000,0000,0000,,value of see any constant we can\Nadd on to the anti derivative
Dialogue: 0,0:06:37.71,0:06:41.30,Default,,0000,0000,0000,,we've found and then we've got\Nanother antiderivative. So in
Dialogue: 0,0:06:41.30,0:06:45.25,Default,,0000,0000,0000,,fact there are lots and lots of\Nantiderivatives of 6X plus
Dialogue: 0,0:06:45.25,0:06:49.20,Default,,0000,0000,0000,,Seven, just as another example,\Nwe could have had over here.
Dialogue: 0,0:06:50.86,0:06:53.94,Default,,0000,0000,0000,,Three X squared plus 7X minus 8.
Dialogue: 0,0:06:55.59,0:07:00.51,Default,,0000,0000,0000,,Or even just three X squared\Nplus 7X where the constant term
Dialogue: 0,0:07:00.51,0:07:05.02,Default,,0000,0000,0000,,was zero. So lots and lots of\Ndifferent antiderivatives for a
Dialogue: 0,0:07:05.02,0:07:06.66,Default,,0000,0000,0000,,single term over here.
Dialogue: 0,0:07:09.12,0:07:10.57,Default,,0000,0000,0000,,Let's do another example.
Dialogue: 0,0:07:13.45,0:07:14.95,Default,,0000,0000,0000,,Suppose this time we look at.
Dialogue: 0,0:07:15.94,0:07:21.22,Default,,0000,0000,0000,,A cubic function that suppose F\Nof X is 4X cubed.
Dialogue: 0,0:07:23.23,0:07:24.100,Default,,0000,0000,0000,,Minus Seven X squared.
Dialogue: 0,0:07:26.44,0:07:27.64,Default,,0000,0000,0000,,Plus 12X
Dialogue: 0,0:07:29.18,0:07:29.93,Default,,0000,0000,0000,,minus 4.
Dialogue: 0,0:07:32.18,0:07:34.77,Default,,0000,0000,0000,,Again, you know how to\Ndifferentiate this. You've
Dialogue: 0,0:07:34.77,0:07:37.04,Default,,0000,0000,0000,,had a lot of practice\Ndifferentiating functions
Dialogue: 0,0:07:37.04,0:07:37.69,Default,,0000,0000,0000,,like this.
Dialogue: 0,0:07:39.22,0:07:42.75,Default,,0000,0000,0000,,So the derivative DF\NDX is going to be.
Dialogue: 0,0:07:43.77,0:07:45.51,Default,,0000,0000,0000,,Three 412 X squared.
Dialogue: 0,0:07:48.71,0:07:52.73,Default,,0000,0000,0000,,The derivative of minus\NSeven X squared is going to
Dialogue: 0,0:07:52.73,0:07:53.94,Default,,0000,0000,0000,,be minus 14X.
Dialogue: 0,0:07:56.54,0:07:58.97,Default,,0000,0000,0000,,And the derivative\Nof 12 X is just 12.
Dialogue: 0,0:08:01.90,0:08:08.38,Default,,0000,0000,0000,,So think of our little F of X as\Nbeing the function 12 X squared
Dialogue: 0,0:08:08.38,0:08:10.11,Default,,0000,0000,0000,,minus 14X at 12.
Dialogue: 0,0:08:12.47,0:08:16.93,Default,,0000,0000,0000,,Notice again the point that\Nthe minus four in the
Dialogue: 0,0:08:16.93,0:08:18.27,Default,,0000,0000,0000,,differentiation process\Ndisappears.
Dialogue: 0,0:08:19.84,0:08:23.47,Default,,0000,0000,0000,,So now ask the question, what's\Nan anti derivative of this
Dialogue: 0,0:08:23.47,0:08:24.79,Default,,0000,0000,0000,,function here little F.
Dialogue: 0,0:08:25.63,0:08:29.10,Default,,0000,0000,0000,,Well, we've got one answer. It's\Non the page already. It's this
Dialogue: 0,0:08:29.10,0:08:31.12,Default,,0000,0000,0000,,big F is an anti derivative of
Dialogue: 0,0:08:31.12,0:08:35.69,Default,,0000,0000,0000,,Little F. But we've seen\Nthat any constant added on
Dialogue: 0,0:08:35.69,0:08:38.38,Default,,0000,0000,0000,,here will still yield\Nanother antiderivative, so
Dialogue: 0,0:08:38.38,0:08:42.24,Default,,0000,0000,0000,,we can write down lots of\Nother ones, for example.
Dialogue: 0,0:08:44.30,0:08:46.26,Default,,0000,0000,0000,,If we add on a constant.
Dialogue: 0,0:08:48.46,0:08:51.45,Default,,0000,0000,0000,,And let's suppose we add on the\Nconstant 10 onto this one.
Dialogue: 0,0:08:52.12,0:08:57.86,Default,,0000,0000,0000,,Then we'll get the function 4X\Ncubed, minus Seven X squared.
Dialogue: 0,0:08:58.80,0:09:03.72,Default,,0000,0000,0000,,Plus 12X plus six is also an\Nanti derivative of Little F.
Dialogue: 0,0:09:04.89,0:09:08.17,Default,,0000,0000,0000,,And the same goes by adding\Nadding on any other constant at
Dialogue: 0,0:09:08.17,0:09:11.17,Default,,0000,0000,0000,,all we choose. If we add on\Nminus six to this.
Dialogue: 0,0:09:14.91,0:09:19.56,Default,,0000,0000,0000,,Will get 4X cubed, minus Seven X\Nsquared plus 12X and adding on
Dialogue: 0,0:09:19.56,0:09:23.86,Default,,0000,0000,0000,,minus six or just leave us this\Nso that that is another
Dialogue: 0,0:09:23.86,0:09:26.01,Default,,0000,0000,0000,,antiderivative. There's an\Ninfinite number of
Dialogue: 0,0:09:26.01,0:09:27.44,Default,,0000,0000,0000,,antiderivatives of Little F.
Dialogue: 0,0:09:28.81,0:09:30.26,Default,,0000,0000,0000,,Now in all the examples we've
Dialogue: 0,0:09:30.26,0:09:34.25,Default,,0000,0000,0000,,looked at. I've in this in a\Nsense, giving you the answer
Dialogue: 0,0:09:34.25,0:09:37.45,Default,,0000,0000,0000,,right at the beginning because\Nwe started with big F, we
Dialogue: 0,0:09:37.45,0:09:41.23,Default,,0000,0000,0000,,differentiated it to find little\NF, so we knew the answer all the
Dialogue: 0,0:09:41.23,0:09:44.73,Default,,0000,0000,0000,,time. In practice, you won't\Nknow the answer all the time, so
Dialogue: 0,0:09:44.73,0:09:47.93,Default,,0000,0000,0000,,one way that we can help\Nourselves by referring to a
Dialogue: 0,0:09:47.93,0:09:50.55,Default,,0000,0000,0000,,table of antiderivatives. Now a\Ntable of antiderivatives will
Dialogue: 0,0:09:50.55,0:09:52.00,Default,,0000,0000,0000,,look something like this one.
Dialogue: 0,0:09:53.00,0:09:56.89,Default,,0000,0000,0000,,There's a copy of this in the\Nnotes accompanying the video and
Dialogue: 0,0:09:56.89,0:09:59.80,Default,,0000,0000,0000,,a table of antiderivatives were\Nlist lots of functions.
Dialogue: 0,0:10:00.16,0:10:00.57,Default,,0000,0000,0000,,F.
Dialogue: 0,0:10:01.72,0:10:03.74,Default,,0000,0000,0000,,And then the anti derivative.
Dialogue: 0,0:10:04.39,0:10:08.25,Default,,0000,0000,0000,,Big F in the next column and\Nyou'll see every anti
Dialogue: 0,0:10:08.25,0:10:12.46,Default,,0000,0000,0000,,derivative will have a plus.\NSee attached to it where C is
Dialogue: 0,0:10:12.46,0:10:14.92,Default,,0000,0000,0000,,an arbitrary constant. Any\Nconstant you choose.
Dialogue: 0,0:10:23.82,0:10:28.71,Default,,0000,0000,0000,,What I want to do now is I\Nwant to link the concept
Dialogue: 0,0:10:28.71,0:10:31.34,Default,,0000,0000,0000,,of an antiderivative with\Nintegration as summation.
Dialogue: 0,0:10:33.21,0:10:36.06,Default,,0000,0000,0000,,Let's think of a function.
Dialogue: 0,0:10:38.96,0:10:40.84,Default,,0000,0000,0000,,Y equals F of X.
Dialogue: 0,0:10:46.04,0:10:48.79,Default,,0000,0000,0000,,Let's suppose the graph of the\Nfunction looks like this.
Dialogue: 0,0:10:51.00,0:10:53.66,Default,,0000,0000,0000,,I'm going to consider\Nfunctions which lie entirely
Dialogue: 0,0:10:53.66,0:10:56.99,Default,,0000,0000,0000,,above the X axis, so I'm\Ngoing to restrict our
Dialogue: 0,0:10:56.99,0:11:00.66,Default,,0000,0000,0000,,attention to the region above\Nthe X axis were looking up
Dialogue: 0,0:11:00.66,0:11:00.99,Default,,0000,0000,0000,,here.
Dialogue: 0,0:11:02.24,0:11:05.91,Default,,0000,0000,0000,,And also I want to restrict\Nattention to the right of the Y
Dialogue: 0,0:11:05.91,0:11:09.01,Default,,0000,0000,0000,,axis, so I'm looking to the\Nright of this line here.
Dialogue: 0,0:11:11.55,0:11:16.75,Default,,0000,0000,0000,,Let's ask ourselves what is the\Narea under the graph of Y equals
Dialogue: 0,0:11:16.75,0:11:17.95,Default,,0000,0000,0000,,F of X?
Dialogue: 0,0:11:19.65,0:11:23.56,Default,,0000,0000,0000,,Well, surely that depends on\Nhow far I want to move to the
Dialogue: 0,0:11:23.56,0:11:26.27,Default,,0000,0000,0000,,right hand side. Let's\Nsuppose I want to move.
Dialogue: 0,0:11:28.06,0:11:31.03,Default,,0000,0000,0000,,To the place where X\Nhas an X Coordinate X.
Dialogue: 0,0:11:33.59,0:11:35.56,Default,,0000,0000,0000,,Then clearly, if X is a large
Dialogue: 0,0:11:35.56,0:11:39.12,Default,,0000,0000,0000,,number. The area under this\Ngraph here will be large,
Dialogue: 0,0:11:39.12,0:11:42.44,Default,,0000,0000,0000,,whereas if X is a small\Nnumber, the area under the
Dialogue: 0,0:11:42.44,0:11:45.46,Default,,0000,0000,0000,,graph will be quite small\Nindeed. Effects is actually 0,
Dialogue: 0,0:11:45.46,0:11:49.39,Default,,0000,0000,0000,,so this line is actually lying\Non the Y axis than the area
Dialogue: 0,0:11:49.39,0:11:51.20,Default,,0000,0000,0000,,under the graph will be 0.
Dialogue: 0,0:11:52.83,0:11:55.97,Default,,0000,0000,0000,,I'm going to do note the\Narea under the graph by a,
Dialogue: 0,0:11:55.97,0:11:57.28,Default,,0000,0000,0000,,so A is the area.
Dialogue: 0,0:12:01.68,0:12:02.27,Default,,0000,0000,0000,,Under
Dialogue: 0,0:12:04.05,0:12:05.24,Default,,0000,0000,0000,,why is F of X?
Dialogue: 0,0:12:07.49,0:12:11.72,Default,,0000,0000,0000,,And as I've just said, the area\Nwill depend upon the value that
Dialogue: 0,0:12:11.72,0:12:15.94,Default,,0000,0000,0000,,we choose for X. In other\Nwords, A is a function of XA
Dialogue: 0,0:12:15.94,0:12:20.82,Default,,0000,0000,0000,,depends upon X, so we write\Nthat like this, a is a of X and
Dialogue: 0,0:12:20.82,0:12:24.72,Default,,0000,0000,0000,,that shows the dependence of\Nthe area on the value we choose
Dialogue: 0,0:12:24.72,0:12:28.94,Default,,0000,0000,0000,,for X. As I said, LG X larger\Narea small acts smaller area.
Dialogue: 0,0:12:30.60,0:12:34.72,Default,,0000,0000,0000,,Ask yourself. What is the height\Nof this line here?
Dialogue: 0,0:12:37.32,0:12:38.93,Default,,0000,0000,0000,,Well, this point here.
Dialogue: 0,0:12:39.59,0:12:42.100,Default,,0000,0000,0000,,Lies. On the curve Y equals F of
Dialogue: 0,0:12:42.100,0:12:48.43,Default,,0000,0000,0000,,X. So the Y value at that point\Nis simply the function evaluated
Dialogue: 0,0:12:48.43,0:12:49.86,Default,,0000,0000,0000,,at this X value.
Dialogue: 0,0:12:50.58,0:12:53.13,Default,,0000,0000,0000,,So the Y value here is just F of
Dialogue: 0,0:12:53.13,0:12:57.44,Default,,0000,0000,0000,,X. In other words, the\Nheight of this line or the
Dialogue: 0,0:12:57.44,0:13:01.14,Default,,0000,0000,0000,,length of this line is just\NF of X. The function
Dialogue: 0,0:13:01.14,0:13:02.48,Default,,0000,0000,0000,,evaluated at this point.
Dialogue: 0,0:13:04.94,0:13:08.68,Default,,0000,0000,0000,,Now I'd like you to think what\Nhappens if we just increase X by
Dialogue: 0,0:13:08.68,0:13:09.75,Default,,0000,0000,0000,,a very small amount.
Dialogue: 0,0:13:10.34,0:13:13.03,Default,,0000,0000,0000,,So I want to just move this\Npoint a little bit further
Dialogue: 0,0:13:13.03,0:13:13.70,Default,,0000,0000,0000,,to the right.
Dialogue: 0,0:13:14.96,0:13:16.12,Default,,0000,0000,0000,,And see what happens.
Dialogue: 0,0:13:17.11,0:13:21.13,Default,,0000,0000,0000,,I'm going to increase X by a\Nlittle bit and that little bit
Dialogue: 0,0:13:21.13,0:13:25.14,Default,,0000,0000,0000,,that distance in here. I'm going\Nto call Delta X. Delta X stands
Dialogue: 0,0:13:25.14,0:13:29.78,Default,,0000,0000,0000,,for a small change in X or we\Ncall it a small increment in X.
Dialogue: 0,0:13:31.93,0:13:34.10,Default,,0000,0000,0000,,By increasing axle little bit.
Dialogue: 0,0:13:34.89,0:13:37.92,Default,,0000,0000,0000,,What we're doing is we're adding\Na little bit more to this area.
Dialogue: 0,0:13:39.72,0:13:42.45,Default,,0000,0000,0000,,This is the additional\Ncontribution to the area. This
Dialogue: 0,0:13:42.45,0:13:43.66,Default,,0000,0000,0000,,shaded region in here.
Dialogue: 0,0:13:44.80,0:13:47.74,Default,,0000,0000,0000,,And that's a little bit extra\Narea, so I'm going to call that.
Dialogue: 0,0:13:49.83,0:13:53.73,Default,,0000,0000,0000,,Delta A, That's an incrementing\Narea changing area.
Dialogue: 0,0:13:55.63,0:13:57.77,Default,,0000,0000,0000,,I'm going to write down\Nan expression for Delta
Dialogue: 0,0:13:57.77,0:13:59.20,Default,,0000,0000,0000,,a try and work it out.
Dialogue: 0,0:14:01.94,0:14:06.00,Default,,0000,0000,0000,,Let's try and see what it is,\Nbut I can't get it exactly. But
Dialogue: 0,0:14:06.00,0:14:09.77,Default,,0000,0000,0000,,if I note that the height of\Nthis line is F of X.
Dialogue: 0,0:14:10.50,0:14:14.66,Default,,0000,0000,0000,,And the width of this column in\Nhere is Delta X. Then I can get
Dialogue: 0,0:14:14.66,0:14:17.70,Default,,0000,0000,0000,,an approximate value for this\Narea by assuming that it's a
Dialogue: 0,0:14:17.70,0:14:20.47,Default,,0000,0000,0000,,rectangular section. In other\Nwords, I'm going to ignore this
Dialogue: 0,0:14:20.47,0:14:22.41,Default,,0000,0000,0000,,little bit at the top in there.
Dialogue: 0,0:14:23.34,0:14:27.14,Default,,0000,0000,0000,,If I assume it's a rectangular\Nsection, then the area here this
Dialogue: 0,0:14:27.14,0:14:28.41,Default,,0000,0000,0000,,additional area Delta A.
Dialogue: 0,0:14:29.32,0:14:32.65,Default,,0000,0000,0000,,Is F of X multiplied by Delta X.
Dialogue: 0,0:14:35.98,0:14:39.03,Default,,0000,0000,0000,,Now, that's only approximately\Ntrue, so this is really an
Dialogue: 0,0:14:39.03,0:14:40.25,Default,,0000,0000,0000,,approximately equal to symbol.
Dialogue: 0,0:14:41.17,0:14:43.49,Default,,0000,0000,0000,,This additional area is\Napproximately equal to
Dialogue: 0,0:14:43.49,0:14:45.47,Default,,0000,0000,0000,,F of X Times Delta X.
Dialogue: 0,0:14:48.21,0:14:53.92,Default,,0000,0000,0000,,Let me now divide both sides by\NDelta X. That's going to give me
Dialogue: 0,0:14:53.92,0:14:58.82,Default,,0000,0000,0000,,Delta a over Delta. X is\Napproximately equal to F of X.
Dialogue: 0,0:15:01.06,0:15:04.24,Default,,0000,0000,0000,,How can we make it more\Naccurate? Well, one way we can
Dialogue: 0,0:15:04.24,0:15:07.42,Default,,0000,0000,0000,,make this more accurate is by\Nchoosing this column to be even
Dialogue: 0,0:15:07.42,0:15:10.07,Default,,0000,0000,0000,,thinner by letting Delta X be\Nsmaller, because then this
Dialogue: 0,0:15:10.07,0:15:12.72,Default,,0000,0000,0000,,additional contribution that\NI've got in here that I didn't
Dialogue: 0,0:15:12.72,0:15:16.43,Default,,0000,0000,0000,,count is reducing its reducing\Nin size. So what I want to do is
Dialogue: 0,0:15:16.43,0:15:20.14,Default,,0000,0000,0000,,I want to let Delta X get even\Nsmaller and smaller and in the
Dialogue: 0,0:15:20.14,0:15:21.100,Default,,0000,0000,0000,,end I want to take the limit.
Dialogue: 0,0:15:22.99,0:15:27.86,Default,,0000,0000,0000,,As Delta X tends to zero of\NDelta over Delta X.
Dialogue: 0,0:15:28.42,0:15:31.44,Default,,0000,0000,0000,,And when I do that, this\Napproximation in here will
Dialogue: 0,0:15:31.44,0:15:34.46,Default,,0000,0000,0000,,become exact and that will give\Nus F of X.
Dialogue: 0,0:15:36.62,0:15:40.67,Default,,0000,0000,0000,,Now, if you've studied the unit\Non differentiation from first
Dialogue: 0,0:15:40.67,0:15:44.72,Default,,0000,0000,0000,,principles, you realize that\Nthis in here is the definition
Dialogue: 0,0:15:44.72,0:15:49.98,Default,,0000,0000,0000,,of the derivative of A with\Nrespect to X, which we write as
Dialogue: 0,0:15:49.98,0:15:55.66,Default,,0000,0000,0000,,DADX. So we have the result that\NDADX is a little F of X.
Dialogue: 0,0:15:58.54,0:16:00.11,Default,,0000,0000,0000,,Let's explore this a\Nlittle bit further.
Dialogue: 0,0:16:03.83,0:16:09.02,Default,,0000,0000,0000,,We have the ADX is a little\NF of X.
Dialogue: 0,0:16:10.48,0:16:13.98,Default,,0000,0000,0000,,What does this mean? Well,\Nit means that little laugh
Dialogue: 0,0:16:13.98,0:16:15.60,Default,,0000,0000,0000,,is the derivative of A.
Dialogue: 0,0:16:26.29,0:16:29.100,Default,,0000,0000,0000,,But it also means if we think\Nback to the discussion that we
Dialogue: 0,0:16:29.100,0:16:33.70,Default,,0000,0000,0000,,had at the beginning of this\Nvideo, that a must be the anti
Dialogue: 0,0:16:33.70,0:16:38.44,Default,,0000,0000,0000,,derivative of F. A is\Nan anti derivative.
Dialogue: 0,0:16:45.74,0:16:53.33,Default,,0000,0000,0000,,F. In other words, the area is\NF of X. The anti derivative of
Dialogue: 0,0:16:53.33,0:16:56.04,Default,,0000,0000,0000,,little F was any constant.
Dialogue: 0,0:16:58.97,0:17:02.64,Default,,0000,0000,0000,,That's going to be an important\Nresult. What it's saying is that
Dialogue: 0,0:17:02.64,0:17:07.54,Default,,0000,0000,0000,,if we have a function, why is F\Nof X and we want to find the
Dialogue: 0,0:17:07.54,0:17:11.21,Default,,0000,0000,0000,,area under the graph? What we do\Nis we calculate an anti
Dialogue: 0,0:17:11.21,0:17:13.05,Default,,0000,0000,0000,,derivative of Little F which is
Dialogue: 0,0:17:13.05,0:17:15.58,Default,,0000,0000,0000,,big F. And use this expression.
Dialogue: 0,0:17:16.20,0:17:17.48,Default,,0000,0000,0000,,To find the area.
Dialogue: 0,0:17:18.52,0:17:21.30,Default,,0000,0000,0000,,There's one thing we don't know\Nin this expression at the
Dialogue: 0,0:17:21.30,0:17:23.58,Default,,0000,0000,0000,,moment, and it's this CC.\NRemember, is an arbitrary
Dialogue: 0,0:17:23.58,0:17:27.12,Default,,0000,0000,0000,,constant, but we can get a value\Nfor. See if we just look back
Dialogue: 0,0:17:27.12,0:17:28.64,Default,,0000,0000,0000,,again. So the graph I drew.
Dialogue: 0,0:17:30.34,0:17:34.37,Default,,0000,0000,0000,,And ask yourself what will be\Nthe area under this curve when X
Dialogue: 0,0:17:34.37,0:17:38.40,Default,,0000,0000,0000,,is chosen to be zero? Well, if\Nyou remember, we said that if
Dialogue: 0,0:17:38.40,0:17:41.50,Default,,0000,0000,0000,,this vertical line here had been\Non the Y axis.
Dialogue: 0,0:17:42.34,0:17:44.49,Default,,0000,0000,0000,,Then the area under the\Ncurve would have been 0.
Dialogue: 0,0:17:45.58,0:17:48.57,Default,,0000,0000,0000,,So this gives us a\Ncondition. It tells us that
Dialogue: 0,0:17:48.57,0:17:50.96,Default,,0000,0000,0000,,when X is 0, the area is 0.
Dialogue: 0,0:17:56.89,0:17:58.76,Default,,0000,0000,0000,,When X is 0.
Dialogue: 0,0:18:00.24,0:18:04.85,Default,,0000,0000,0000,,Area is 0. What does this mean?\NWhile the area being 0?
Dialogue: 0,0:18:06.55,0:18:07.86,Default,,0000,0000,0000,,X being 0.
Dialogue: 0,0:18:09.72,0:18:13.68,Default,,0000,0000,0000,,Plus a constant and this\Ncondition then gives us a value
Dialogue: 0,0:18:13.68,0:18:18.00,Default,,0000,0000,0000,,for C, so C must be equal to\Nminus F of North.
Dialogue: 0,0:18:20.69,0:18:26.02,Default,,0000,0000,0000,,And that value for C and go back\Nin this result here. So we have
Dialogue: 0,0:18:26.02,0:18:30.63,Default,,0000,0000,0000,,the final result that the area\Nunder the graph is given by big
Dialogue: 0,0:18:30.63,0:18:33.47,Default,,0000,0000,0000,,F of X minus big F of note.
Dialogue: 0,0:18:40.81,0:18:42.36,Default,,0000,0000,0000,,Let me just write that\Ndown again.
Dialogue: 0,0:18:51.14,0:18:51.56,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:18:55.67,0:18:59.86,Default,,0000,0000,0000,,Now let's look at this problem.\NSupposing that I'm interested in
Dialogue: 0,0:18:59.86,0:19:04.81,Default,,0000,0000,0000,,finding the area under the graph\Nof Y is little F of X.
Dialogue: 0,0:19:08.46,0:19:09.63,Default,,0000,0000,0000,,Up to the point.
Dialogue: 0,0:19:11.53,0:19:13.70,Default,,0000,0000,0000,,Where X has the value be.
Dialogue: 0,0:19:14.64,0:19:19.74,Default,,0000,0000,0000,,So I want the area from the Y\Naxis, which is where we're
Dialogue: 0,0:19:19.74,0:19:23.66,Default,,0000,0000,0000,,working from above the X axis up\Nto the point.
Dialogue: 0,0:19:24.28,0:19:25.48,Default,,0000,0000,0000,,Where X equals B.
Dialogue: 0,0:19:26.48,0:19:28.65,Default,,0000,0000,0000,,We can use this boxed\Nformula here.
Dialogue: 0,0:19:30.07,0:19:34.45,Default,,0000,0000,0000,,When X is be will get a of B.\NThat's the area up to be.
Dialogue: 0,0:19:36.50,0:19:38.63,Default,,0000,0000,0000,,Will be F of B.
Dialogue: 0,0:19:41.32,0:19:43.94,Default,,0000,0000,0000,,Minus F of note.
Dialogue: 0,0:19:45.42,0:19:48.91,Default,,0000,0000,0000,,So this expression will give you\Nthe area up to be.
Dialogue: 0,0:19:58.85,0:20:00.11,Default,,0000,0000,0000,,Suppose now I want.
Dialogue: 0,0:20:00.18,0:20:03.74,Default,,0000,0000,0000,,Area up to A and let's\NSuppose A is about here.
Dialogue: 0,0:20:04.84,0:20:06.01,Default,,0000,0000,0000,,So now I'm interested.
Dialogue: 0,0:20:08.08,0:20:13.36,Default,,0000,0000,0000,,In this area in here, which is\Nthe area from the Y axis up to a
Dialogue: 0,0:20:13.36,0:20:17.32,Default,,0000,0000,0000,,well, again using the same\Nformula, the area up to a which
Dialogue: 0,0:20:17.32,0:20:18.64,Default,,0000,0000,0000,,is a of A.
Dialogue: 0,0:20:20.15,0:20:23.77,Default,,0000,0000,0000,,Is F evaluated at the X value\Nwhich is a?
Dialogue: 0,0:20:26.47,0:20:29.33,Default,,0000,0000,0000,,Subtract. Big F of note.
Dialogue: 0,0:20:31.77,0:20:34.63,Default,,0000,0000,0000,,So I have two expressions on\Nthe page here, one for the
Dialogue: 0,0:20:34.63,0:20:35.58,Default,,0000,0000,0000,,area up to be.
Dialogue: 0,0:20:37.36,0:20:38.94,Default,,0000,0000,0000,,And one for the area up to A.
Dialogue: 0,0:20:40.66,0:20:45.76,Default,,0000,0000,0000,,Ask yourself How do I find\Nthis area in here? That's
Dialogue: 0,0:20:45.76,0:20:47.62,Default,,0000,0000,0000,,the area between A&B.
Dialogue: 0,0:20:49.16,0:20:54.28,Default,,0000,0000,0000,,Well, the area between A&B we\Ncan think of as the area up to
Dialogue: 0,0:20:54.28,0:20:57.58,Default,,0000,0000,0000,,be. Subtract the area up to A.
Dialogue: 0,0:20:58.82,0:21:02.58,Default,,0000,0000,0000,,So if we find the difference of\Nthese two quantities, that will
Dialogue: 0,0:21:02.58,0:21:04.45,Default,,0000,0000,0000,,give us the area between A&B.
Dialogue: 0,0:21:05.33,0:21:06.79,Default,,0000,0000,0000,,So we get the area.
Dialogue: 0,0:21:09.73,0:21:11.91,Default,,0000,0000,0000,,Under Wise F of X.
Dialogue: 0,0:21:14.96,0:21:18.78,Default,,0000,0000,0000,,From X equals A to X equals
Dialogue: 0,0:21:18.78,0:21:23.46,Default,,0000,0000,0000,,B. Is the area under\Nthis graph from X is A
Dialogue: 0,0:21:23.46,0:21:25.85,Default,,0000,0000,0000,,to X is B is given by?
Dialogue: 0,0:21:26.89,0:21:29.90,Default,,0000,0000,0000,,Once the area up to be, subtract\Nthe area up to A.
Dialogue: 0,0:21:30.52,0:21:33.19,Default,,0000,0000,0000,,So if we just find the\Ndifference of these two
Dialogue: 0,0:21:33.19,0:21:35.86,Default,,0000,0000,0000,,quantities will have F of B\Nminus F of A.
Dialogue: 0,0:21:40.81,0:21:45.22,Default,,0000,0000,0000,,Add minus F of not minus minus F\Nof notes. These terms will
Dialogue: 0,0:21:45.22,0:21:49.28,Default,,0000,0000,0000,,cancel out. So In other words\Nthat's the result we need. The
Dialogue: 0,0:21:49.28,0:21:51.100,Default,,0000,0000,0000,,area under this graph between\NA&B is just.
Dialogue: 0,0:21:52.61,0:21:54.81,Default,,0000,0000,0000,,Big F of B minus big FA.
Dialogue: 0,0:21:59.57,0:22:01.52,Default,,0000,0000,0000,,Let me just write that\Ndown again.
Dialogue: 0,0:22:26.38,0:22:30.89,Default,,0000,0000,0000,,The area and Y equals F of X\Nbetween ex is an ex is B is
Dialogue: 0,0:22:30.89,0:22:35.69,Default,,0000,0000,0000,,given by big F of B minus big F\Nof a where big F remember is an
Dialogue: 0,0:22:35.69,0:22:37.10,Default,,0000,0000,0000,,anti derivative of Little F.
Dialogue: 0,0:22:42.05,0:22:43.30,Default,,0000,0000,0000,,Any answer derivative?
Dialogue: 0,0:22:54.45,0:22:57.57,Default,,0000,0000,0000,,So this is a very important\Nresult because it means that if
Dialogue: 0,0:22:57.57,0:22:59.39,Default,,0000,0000,0000,,I give you a function little F
Dialogue: 0,0:22:59.39,0:23:04.12,Default,,0000,0000,0000,,of X. And you can calculate an\Nanti derivative of its big F.
Dialogue: 0,0:23:04.82,0:23:08.47,Default,,0000,0000,0000,,And you can find the area\Nunder the graph merely by
Dialogue: 0,0:23:08.47,0:23:10.80,Default,,0000,0000,0000,,evaluating that anti\Nderivative at B, evaluating
Dialogue: 0,0:23:10.80,0:23:13.78,Default,,0000,0000,0000,,it A and finding the\Ndifference of the two
Dialogue: 0,0:23:13.78,0:23:14.12,Default,,0000,0000,0000,,quantities.
Dialogue: 0,0:23:16.47,0:23:19.59,Default,,0000,0000,0000,,Now in the previous video on\Nintegration by summation, the
Dialogue: 0,0:23:19.59,0:23:23.02,Default,,0000,0000,0000,,area under under a graph was\Nfound in a slightly different
Dialogue: 0,0:23:23.02,0:23:24.58,Default,,0000,0000,0000,,way. What was done there?
Dialogue: 0,0:23:25.20,0:23:30.88,Default,,0000,0000,0000,,Was that the area under the\Ngraph between A&B was found by
Dialogue: 0,0:23:30.88,0:23:35.13,Default,,0000,0000,0000,,dividing the area into lots of\Nthin rectangular strips?
Dialogue: 0,0:23:36.33,0:23:39.58,Default,,0000,0000,0000,,Finding the area of each of the\Nrectangles and adding them all
Dialogue: 0,0:23:39.58,0:23:45.31,Default,,0000,0000,0000,,up. And doing that we had this\Nresult that the area under the
Dialogue: 0,0:23:45.31,0:23:50.20,Default,,0000,0000,0000,,graph between A&B was given by\Nthe limit as Delta X tends to
Dialogue: 0,0:23:50.20,0:23:54.71,Default,,0000,0000,0000,,zero of the sum of all these\Nrectangular areas, which was F
Dialogue: 0,0:23:54.71,0:24:00.73,Default,,0000,0000,0000,,of X Delta X between X is A and\NX is be. If you're not familiar
Dialogue: 0,0:24:00.73,0:24:05.99,Default,,0000,0000,0000,,with that, I would advise you go\Nback and have a look at that
Dialogue: 0,0:24:05.99,0:24:09.38,Default,,0000,0000,0000,,other video on integration by\Nsummation, but that formula.
Dialogue: 0,0:24:10.20,0:24:13.14,Default,,0000,0000,0000,,For this area was derived\Nin that video.
Dialogue: 0,0:24:14.19,0:24:17.54,Default,,0000,0000,0000,,And this formula\Ndefines what we mean
Dialogue: 0,0:24:17.54,0:24:21.36,Default,,0000,0000,0000,,by the definite\Nintegral from A to B
Dialogue: 0,0:24:21.36,0:24:23.75,Default,,0000,0000,0000,,of F of X DX.
Dialogue: 0,0:24:25.40,0:24:28.13,Default,,0000,0000,0000,,That's how we defined\Na definite integral.
Dialogue: 0,0:24:29.40,0:24:32.44,Default,,0000,0000,0000,,And if we wanted to define a\Ndefinite integral, if we wanted
Dialogue: 0,0:24:32.44,0:24:35.72,Default,,0000,0000,0000,,to calculate a definite integral\Nin that video, we had to do it
Dialogue: 0,0:24:35.72,0:24:38.26,Default,,0000,0000,0000,,through the process of finding\Nthe limit of a sum.
Dialogue: 0,0:24:39.78,0:24:43.78,Default,,0000,0000,0000,,Now the the process of finding\Nthe limit of a sum is quite
Dialogue: 0,0:24:43.78,0:24:47.26,Default,,0000,0000,0000,,cumbersome and impractical. But\Nwhat we've learned now is that
Dialogue: 0,0:24:47.26,0:24:51.43,Default,,0000,0000,0000,,we don't have to use the limit\Nof a sum to find the area. We
Dialogue: 0,0:24:51.43,0:24:52.82,Default,,0000,0000,0000,,can use these anti derivatives.
Dialogue: 0,0:24:54.12,0:24:56.91,Default,,0000,0000,0000,,Because we've got two\Nexpressions for the area, we've
Dialogue: 0,0:24:56.91,0:25:00.57,Default,,0000,0000,0000,,got this expression. As a\Ndefinite integral, and we've got
Dialogue: 0,0:25:00.57,0:25:03.38,Default,,0000,0000,0000,,this expression in terms of\Nantiderivatives, and if we put
Dialogue: 0,0:25:03.38,0:25:05.06,Default,,0000,0000,0000,,all that together will end up
Dialogue: 0,0:25:05.06,0:25:09.06,Default,,0000,0000,0000,,with this result. The definite\Nintegral from A to B.
Dialogue: 0,0:25:10.43,0:25:11.81,Default,,0000,0000,0000,,F of X DX.
Dialogue: 0,0:25:13.18,0:25:14.47,Default,,0000,0000,0000,,Is FB.
Dialogue: 0,0:25:16.01,0:25:17.90,Default,,0000,0000,0000,,Minus F of A.
Dialogue: 0,0:25:21.86,0:25:26.34,Default,,0000,0000,0000,,In other words, the definite\Nintegral of little F between the
Dialogue: 0,0:25:26.34,0:25:30.00,Default,,0000,0000,0000,,limits of A&B is found by\Nevaluating this expression,
Dialogue: 0,0:25:30.00,0:25:34.07,Default,,0000,0000,0000,,where big F is an anti\Nderivative of Little F.
Dialogue: 0,0:25:39.80,0:25:41.61,Default,,0000,0000,0000,,Let me write that formula\Ndown again.
Dialogue: 0,0:25:43.27,0:25:45.21,Default,,0000,0000,0000,,The integral from A to B.
Dialogue: 0,0:25:46.71,0:25:54.09,Default,,0000,0000,0000,,F of X DX is given by\Nbig FB minus big F of A.
Dialogue: 0,0:25:55.86,0:25:56.95,Default,,0000,0000,0000,,Let me give you an example.
Dialogue: 0,0:26:02.13,0:26:06.01,Default,,0000,0000,0000,,Suppose we're interested\Nin the problem of finding
Dialogue: 0,0:26:06.01,0:26:10.86,Default,,0000,0000,0000,,the area under the graph\Nof Y equals X squared.
Dialogue: 0,0:26:14.52,0:26:17.35,Default,,0000,0000,0000,,And let's suppose for the sake\Nof argument we want the area
Dialogue: 0,0:26:17.35,0:26:21.82,Default,,0000,0000,0000,,under the graph. Between X is\Nnot an ex is one, so we're
Dialogue: 0,0:26:21.82,0:26:23.02,Default,,0000,0000,0000,,interested in this area.
Dialogue: 0,0:26:26.03,0:26:29.59,Default,,0000,0000,0000,,In the previous unit on\Nintegration as a summation,
Dialogue: 0,0:26:29.59,0:26:33.95,Default,,0000,0000,0000,,this was done by dividing\Nthis area into lots of thin
Dialogue: 0,0:26:33.95,0:26:34.74,Default,,0000,0000,0000,,rectangular strips.
Dialogue: 0,0:26:36.04,0:26:40.36,Default,,0000,0000,0000,,And finding the area of each of\Nthose rectangles separately.
Dialogue: 0,0:26:40.98,0:26:42.19,Default,,0000,0000,0000,,And then adding them all up.
Dialogue: 0,0:26:43.08,0:26:47.58,Default,,0000,0000,0000,,That gave rise to this formula\Nthat the area is the limit.
Dialogue: 0,0:26:48.66,0:26:50.75,Default,,0000,0000,0000,,As Delta X tends to 0.
Dialogue: 0,0:26:51.90,0:26:57.26,Default,,0000,0000,0000,,Of the sum from X equals not to\NX equals 1.
Dialogue: 0,0:26:57.94,0:27:00.27,Default,,0000,0000,0000,,Of X squared Delta X.
Dialogue: 0,0:27:01.46,0:27:04.77,Default,,0000,0000,0000,,So the area was expressed as the\Nlimit of a sum.
Dialogue: 0,0:27:05.83,0:27:11.79,Default,,0000,0000,0000,,And in turn, that defines the\Ndefinite integral. X is not to
Dialogue: 0,0:27:11.79,0:27:14.28,Default,,0000,0000,0000,,one of X squared DX.
Dialogue: 0,0:27:16.36,0:27:20.38,Default,,0000,0000,0000,,Now open till now if you\Nwanted to work this area out
Dialogue: 0,0:27:20.38,0:27:24.74,Default,,0000,0000,0000,,the way you would do it would\Nbe by finding the limit of
Dialogue: 0,0:27:24.74,0:27:27.08,Default,,0000,0000,0000,,this some, but that's\Nimpractical and cumbersome.
Dialogue: 0,0:27:27.08,0:27:29.76,Default,,0000,0000,0000,,Instead, we're going to use\Nthis result using
Dialogue: 0,0:27:29.76,0:27:30.10,Default,,0000,0000,0000,,antiderivatives.
Dialogue: 0,0:27:31.78,0:27:35.34,Default,,0000,0000,0000,,Our little F of X in this\Ncase is X squared.
Dialogue: 0,0:27:40.35,0:27:44.51,Default,,0000,0000,0000,,And this formula at the top of\Nthe page tells us that we can
Dialogue: 0,0:27:44.51,0:27:46.59,Default,,0000,0000,0000,,evaluate this definite\Nintegral by finding an
Dialogue: 0,0:27:46.59,0:27:49.56,Default,,0000,0000,0000,,antiderivative capital F. So\Nwe want to do that first.
Dialogue: 0,0:27:49.56,0:27:53.42,Default,,0000,0000,0000,,Well, if little F is X\Nsquared, big F of X, well we
Dialogue: 0,0:27:53.42,0:27:56.98,Default,,0000,0000,0000,,want an anti derivative of X\Nsquared and if you don't know
Dialogue: 0,0:27:56.98,0:27:59.66,Default,,0000,0000,0000,,what one is, you refer back to\Nyour table.
Dialogue: 0,0:28:00.94,0:28:04.69,Default,,0000,0000,0000,,An anti derivative of X\Nsquared is X cubed over 3
Dialogue: 0,0:28:04.69,0:28:05.71,Default,,0000,0000,0000,,plus a constant.
Dialogue: 0,0:28:14.25,0:28:18.63,Default,,0000,0000,0000,,So in order to calculate this\Ndefinite integral, what we need
Dialogue: 0,0:28:18.63,0:28:20.22,Default,,0000,0000,0000,,to do is evaluate.
Dialogue: 0,0:28:21.22,0:28:25.27,Default,,0000,0000,0000,,The Anti Derivative Capital F.\NAt B. That's the upper limit,
Dialogue: 0,0:28:25.27,0:28:27.48,Default,,0000,0000,0000,,which in this case is one.
Dialogue: 0,0:28:28.72,0:28:32.49,Default,,0000,0000,0000,,And then at the lower limit,\Nwhich in the in our case is
Dialogue: 0,0:28:32.49,0:28:34.23,Default,,0000,0000,0000,,zero. Let's workout F of one.
Dialogue: 0,0:28:35.92,0:28:37.86,Default,,0000,0000,0000,,Well, F of one is going to be 1
Dialogue: 0,0:28:37.86,0:28:40.65,Default,,0000,0000,0000,,cubed. Over 3, which is\Njust a third.
Dialogue: 0,0:28:42.11,0:28:42.79,Default,,0000,0000,0000,,Plus C.
Dialogue: 0,0:28:45.17,0:28:46.93,Default,,0000,0000,0000,,Let's workout big F of 0.
Dialogue: 0,0:28:48.34,0:28:49.56,Default,,0000,0000,0000,,Big F of 0.
Dialogue: 0,0:28:50.62,0:28:54.32,Default,,0000,0000,0000,,Is going to be 0 cubed over\Nthree, which is 0 plus. See, so
Dialogue: 0,0:28:54.32,0:28:57.51,Default,,0000,0000,0000,,it's just see. And then we\Nwant to find the difference.
Dialogue: 0,0:28:58.77,0:29:01.28,Default,,0000,0000,0000,,Half of 1 minus F of note.
Dialogue: 0,0:29:02.62,0:29:06.15,Default,,0000,0000,0000,,Will be 1/3 plus C minus C, so\Nthe Seas will cancel and would
Dialogue: 0,0:29:06.15,0:29:07.66,Default,,0000,0000,0000,,be just left with the third.
Dialogue: 0,0:29:08.89,0:29:11.93,Default,,0000,0000,0000,,In other words, to calculate\Nthis integral here.
Dialogue: 0,0:29:13.28,0:29:16.38,Default,,0000,0000,0000,,All we have to do is find the\NAnti Derivative Capital F.
Dialogue: 0,0:29:17.21,0:29:21.24,Default,,0000,0000,0000,,Evaluate it at the upper limit\Nevaluated at the lower limit.
Dialogue: 0,0:29:21.24,0:29:25.26,Default,,0000,0000,0000,,Find the difference and the\Nresult is the third. That's the
Dialogue: 0,0:29:25.26,0:29:28.19,Default,,0000,0000,0000,,area under this graph between\NNorth and one.
Dialogue: 0,0:29:29.66,0:29:32.17,Default,,0000,0000,0000,,Now let me just show you how we\Nwould normally set this out.
Dialogue: 0,0:29:33.20,0:29:34.90,Default,,0000,0000,0000,,We would normally set this\Nout like this.
Dialogue: 0,0:29:38.68,0:29:43.92,Default,,0000,0000,0000,,We find. An anti derivative of X\Nsquared, which we've seen is X
Dialogue: 0,0:29:43.92,0:29:45.56,Default,,0000,0000,0000,,cubed over 3 plus C.
Dialogue: 0,0:29:49.30,0:29:52.67,Default,,0000,0000,0000,,We don't normally write the Plus\NC down, and the reason for that
Dialogue: 0,0:29:52.67,0:29:55.26,Default,,0000,0000,0000,,is when we're finding definite\Nintegrals, the sea will always
Dialogue: 0,0:29:55.26,0:29:58.62,Default,,0000,0000,0000,,cancel out as we saw here, the\NSeas cancelled out in here and
Dialogue: 0,0:29:58.62,0:30:00.18,Default,,0000,0000,0000,,that will always be the case.
Dialogue: 0,0:30:00.38,0:30:03.50,Default,,0000,0000,0000,,So we don't actually need to\Nwrite a plus C down when we
Dialogue: 0,0:30:03.50,0:30:05.42,Default,,0000,0000,0000,,write down an anti\Nderivative of X squared.
Dialogue: 0,0:30:06.59,0:30:10.37,Default,,0000,0000,0000,,It's conventional to write down\Nthe anti derivative in square
Dialogue: 0,0:30:10.37,0:30:14.60,Default,,0000,0000,0000,,brackets. And to transfer the\Nlimits on the original integral,
Dialogue: 0,0:30:14.60,0:30:18.44,Default,,0000,0000,0000,,the Norton one to the right hand\Nside over there like so.
Dialogue: 0,0:30:20.48,0:30:23.91,Default,,0000,0000,0000,,What we then want to do is\Nevaluate the anti derivative at
Dialogue: 0,0:30:23.91,0:30:28.20,Default,,0000,0000,0000,,the top limit that corresponded\Nto the FB or the F of one. So we
Dialogue: 0,0:30:28.20,0:30:31.92,Default,,0000,0000,0000,,work this out at the top limit\Nwhich is 1 cubed over 3.
Dialogue: 0,0:30:35.41,0:30:39.23,Default,,0000,0000,0000,,We work it out at the bottom\Nlimit. That's the F of a or
Dialogue: 0,0:30:39.23,0:30:40.32,Default,,0000,0000,0000,,the F of Norte.
Dialogue: 0,0:30:41.38,0:30:44.84,Default,,0000,0000,0000,,We work this out of the lower\Nlimits will just get zero and
Dialogue: 0,0:30:44.84,0:30:46.97,Default,,0000,0000,0000,,then we want the difference\Nbetween the two.
Dialogue: 0,0:30:48.43,0:30:51.67,Default,,0000,0000,0000,,Which is just going to give\Nus a third. So that's the
Dialogue: 0,0:30:51.67,0:30:54.10,Default,,0000,0000,0000,,normal way we would set out a\Ndefinite integral.
Dialogue: 0,0:31:00.19,0:31:01.45,Default,,0000,0000,0000,,So as we've seen.
Dialogue: 0,0:31:02.76,0:31:07.21,Default,,0000,0000,0000,,Definite integration is very\Nclosely associated with Anti
Dialogue: 0,0:31:07.21,0:31:07.76,Default,,0000,0000,0000,,Differentiation.
Dialogue: 0,0:31:10.19,0:31:14.33,Default,,0000,0000,0000,,Because of this, in general it's\Nuseful to think of Anti
Dialogue: 0,0:31:14.33,0:31:18.09,Default,,0000,0000,0000,,Differentiation as integration\Nand then we would often refer to
Dialogue: 0,0:31:18.09,0:31:21.85,Default,,0000,0000,0000,,a table like this one of\Nantiderivatives as simply a
Dialogue: 0,0:31:21.85,0:31:22.97,Default,,0000,0000,0000,,table of integrals.
Dialogue: 0,0:31:24.02,0:31:27.15,Default,,0000,0000,0000,,And there will be a table of\Nintegrals in the notes and in
Dialogue: 0,0:31:27.15,0:31:30.04,Default,,0000,0000,0000,,later videos you'll be looking\Nin much more detail at how to
Dialogue: 0,0:31:30.04,0:31:32.94,Default,,0000,0000,0000,,use a table of integrals. So\Nyou think of the table of
Dialogue: 0,0:31:32.94,0:31:35.11,Default,,0000,0000,0000,,integrals as your table of\Nantiderivatives and vice versa.
Dialogue: 0,0:31:36.73,0:31:40.26,Default,,0000,0000,0000,,Very early on in the video, we\Nlooked at this problem. We
Dialogue: 0,0:31:40.26,0:31:44.08,Default,,0000,0000,0000,,started off with little F of X\Nbeing equal to four X cubed.
Dialogue: 0,0:31:45.35,0:31:47.20,Default,,0000,0000,0000,,Minus Seven X squared.
Dialogue: 0,0:31:48.28,0:31:51.31,Default,,0000,0000,0000,,Plus 12X minus 4.
Dialogue: 0,0:31:52.39,0:31:53.74,Default,,0000,0000,0000,,We differentiate it.
Dialogue: 0,0:31:57.25,0:32:02.09,Default,,0000,0000,0000,,To give 12 X squared\Nminus 14X plus 12.
Dialogue: 0,0:32:03.14,0:32:08.25,Default,,0000,0000,0000,,And then we said that an anti\Nderivative of 12 X squared minus
Dialogue: 0,0:32:08.25,0:32:09.43,Default,,0000,0000,0000,,14X plus 12.
Dialogue: 0,0:32:10.04,0:32:12.64,Default,,0000,0000,0000,,Was this function\Ncapital F plus C?
Dialogue: 0,0:32:13.84,0:32:17.40,Default,,0000,0000,0000,,We've got a notation we\Ncan use now because we've
Dialogue: 0,0:32:17.40,0:32:19.89,Default,,0000,0000,0000,,linked antiderivatives to\Nintegrals and the notation
Dialogue: 0,0:32:19.89,0:32:23.81,Default,,0000,0000,0000,,we would use is that the\Nintegral of 12 X squared.
Dialogue: 0,0:32:25.26,0:32:31.19,Default,,0000,0000,0000,,Minus 14X plus 12 DX\Nis equal to.
Dialogue: 0,0:32:31.89,0:32:33.25,Default,,0000,0000,0000,,4X cubed
Dialogue: 0,0:32:34.41,0:32:40.18,Default,,0000,0000,0000,,minus Seven X squared plus 12X\Nplus any arbitrary constant.
Dialogue: 0,0:32:41.05,0:32:43.68,Default,,0000,0000,0000,,And we call this an\Nindefinite integral.
Dialogue: 0,0:32:50.12,0:32:54.68,Default,,0000,0000,0000,,And we say that the indefinite\Nintegral of 12 X squared minus
Dialogue: 0,0:32:54.68,0:33:00.00,Default,,0000,0000,0000,,14X plus 12 with respect to X is\N4X cubed minus Seven X squared
Dialogue: 0,0:33:00.00,0:33:02.74,Default,,0000,0000,0000,,plus 12X. Plus a constant\Nof integration.
Dialogue: 0,0:33:06.52,0:33:10.57,Default,,0000,0000,0000,,OK, In summary, what have\Nwe found? What we found
Dialogue: 0,0:33:10.57,0:33:12.19,Default,,0000,0000,0000,,that a definite integral?
Dialogue: 0,0:33:14.09,0:33:20.08,Default,,0000,0000,0000,,The integral from A to B of\Nlittle F of X DX. We found that
Dialogue: 0,0:33:20.08,0:33:21.67,Default,,0000,0000,0000,,this is a number.
Dialogue: 0,0:33:22.53,0:33:28.05,Default,,0000,0000,0000,,And it's obtained from the\Nformula F of B minus FA,
Dialogue: 0,0:33:28.05,0:33:33.07,Default,,0000,0000,0000,,where big F is any anti\Nderivative of Little F.
Dialogue: 0,0:33:34.50,0:33:38.45,Default,,0000,0000,0000,,And we've also seen the\Nindefinite integral of F
Dialogue: 0,0:33:38.45,0:33:39.77,Default,,0000,0000,0000,,of X DX.
Dialogue: 0,0:33:40.88,0:33:46.50,Default,,0000,0000,0000,,Is a function big F of X plus an\Narbitrary constant? Again, where
Dialogue: 0,0:33:46.50,0:33:51.25,Default,,0000,0000,0000,,F is any anti derivative of\NLittle F&C is an arbitrary
Dialogue: 0,0:33:51.25,0:33:56.06,Default,,0000,0000,0000,,constant. So in this video we've\Nlearned how to do
Dialogue: 0,0:33:56.06,0:33:58.58,Default,,0000,0000,0000,,differentiation in reverse.\NWe've learned about
Dialogue: 0,0:33:58.58,0:34:00.48,Default,,0000,0000,0000,,antiderivatives. Definite
Dialogue: 0,0:34:00.48,0:34:04.13,Default,,0000,0000,0000,,integrals. And indefinite\Nintegrals in subsequent videos.
Dialogue: 0,0:34:04.13,0:34:07.77,Default,,0000,0000,0000,,You learn a lot more teak\Ntechniques of integration.