1 00:00:00,470 --> 00:00:01,340 ‫Welcome back. 2 00:00:02,270 --> 00:00:09,310 ‫Now, we have been using a lot the word integration when we compute a new state. 3 00:00:09,740 --> 00:00:17,750 ‫So let's take a random state, let's say Phi Phi, Sub K plus one equals. 4 00:00:18,640 --> 00:00:30,790 ‫Five up subquery plus five Datsyuk times the sample time interval divided by RN, and we're saying that 5 00:00:30,790 --> 00:00:32,530 ‫we are integrating the state. 6 00:00:32,680 --> 00:00:34,370 ‫So why are we saying that? 7 00:00:35,180 --> 00:00:43,540 ‫Well, one way to think about it is that integration is finding an area under the curve. 8 00:00:44,800 --> 00:00:49,630 ‫So I have a graph here and here I have time. 9 00:00:50,680 --> 00:00:56,530 ‫And on this vertical axis I have five dots like this. 10 00:00:57,590 --> 00:01:00,470 ‫And then I have some kind of graph here for that. 11 00:01:01,490 --> 00:01:06,380 ‫So if the vertical axis is five dot, then. 12 00:01:07,400 --> 00:01:15,710 ‫Let's say that I have computed the area under the curve until this point here. 13 00:01:16,900 --> 00:01:20,480 ‫So what would be this white area? 14 00:01:21,520 --> 00:01:22,930 ‫What is its meaning? 15 00:01:24,040 --> 00:01:32,920 ‫Well, this white area is like is this and now let's take a new intervale. 16 00:01:33,880 --> 00:01:41,750 ‫Let's take this interval here in red, which is our PT's over again. 17 00:01:42,520 --> 00:01:45,940 ‫So this one this point here. 18 00:01:47,020 --> 00:01:55,150 ‫It's our FYE dot, OK, so it's this one here, that's this point. 19 00:01:56,160 --> 00:02:02,340 ‫So if you multiply five dot K times, T is over N. 20 00:02:03,400 --> 00:02:14,380 ‫Then this term here, it will be an approximated area under the curve, but only in this region in this 21 00:02:14,380 --> 00:02:16,120 ‫interval over and. 22 00:02:17,190 --> 00:02:26,730 ‫And of course, in reality, you want this interval to be small so that this error that you have over 23 00:02:26,730 --> 00:02:35,730 ‫here, this blue, this small region that this red area is not covering, so that it would be as small 24 00:02:35,730 --> 00:02:36,510 ‫as possible. 25 00:02:37,640 --> 00:02:49,340 ‫And so that means that your new state, fi, so K plus one equals your white area under the curve. 26 00:02:50,460 --> 00:02:55,730 ‫Plus, your red area under the curve, so you're fired. 27 00:02:55,980 --> 00:03:02,880 ‫Plus one would incorporate this and this. 28 00:03:04,080 --> 00:03:15,300 ‫And it makes sense, right, because you fight that Kay is your DFI deti, which essentially is your 29 00:03:15,570 --> 00:03:24,960 ‫delta fly over Delta T is just this is the differential case, the infinitesimally small case. 30 00:03:25,920 --> 00:03:34,500 ‫And this would be like an average slope and then it's over n well, it's a time interval, right. 31 00:03:34,530 --> 00:03:45,450 ‫So it's like Delta T. And so if you multiply this slope by Delta T, then they cancel out and then you 32 00:03:45,450 --> 00:03:50,420 ‫get your Delta Phi and so your fly. 33 00:03:50,550 --> 00:03:57,620 ‫So K plus one equals your fly up K plus Delta Phi. 34 00:03:58,680 --> 00:04:01,020 ‫So that's why it's an integration process. 35 00:04:01,980 --> 00:04:08,790 ‫Now that explanation actually differs from the previous course in this core series, but it's the same 36 00:04:08,790 --> 00:04:09,300 ‫thing. 37 00:04:09,540 --> 00:04:11,070 ‫Just explain differently. 38 00:04:11,970 --> 00:04:19,890 ‫In the previous course I said that you have time on the horizontal axis and then you have an actual 39 00:04:19,890 --> 00:04:21,910 ‫state on the vertical axis. 40 00:04:22,200 --> 00:04:30,180 ‫So now you have PHY on the vertical axis, not five dot, but actual FYE. 41 00:04:31,250 --> 00:04:40,460 ‫So now the area under the curve from the previous explanation is represented on the vertical axis. 42 00:04:41,820 --> 00:04:53,700 ‫So the number that you get here as an area under the curve is now here on the Axis five K, that would 43 00:04:53,700 --> 00:04:54,840 ‫be at this point here. 44 00:04:55,950 --> 00:05:03,300 ‫Because that's actually the main logic of integration in general, when you're computing the area under 45 00:05:03,300 --> 00:05:10,650 ‫the curve, when you take an integral of something, then what you're doing, you're creating some kind 46 00:05:10,650 --> 00:05:17,760 ‫of function where you keep your domain the same, which in this case is time. 47 00:05:18,720 --> 00:05:27,650 ‫But then the area under the curve, you mapped that on a vertical axis and it's no different here, 48 00:05:28,590 --> 00:05:38,190 ‫and if you find that sulphuric here was on a vertical axis, then in this version, it's a slope. 49 00:05:38,910 --> 00:05:40,350 ‫It's the slope of the graph. 50 00:05:41,450 --> 00:05:52,910 ‫So it's over, GNR was our time interval, so if you go from here up until you reach this Red Point, 51 00:05:52,910 --> 00:06:03,820 ‫which represents this slope, but the slope at this point here at K, not that K plus one, but at K. 52 00:06:04,100 --> 00:06:05,530 ‫So that's the slope here. 53 00:06:05,810 --> 00:06:09,800 ‫But then you projected one time interval ahead. 54 00:06:10,790 --> 00:06:17,300 ‫So once you reach this point here, this point on the straight red line. 55 00:06:18,360 --> 00:06:29,640 ‫Then that will be your face of Kate plus one, so this point, that means that their difference here 56 00:06:30,810 --> 00:06:31,800 ‫is Delta. 57 00:06:31,830 --> 00:06:34,160 ‫Why is this now? 58 00:06:35,250 --> 00:06:42,960 ‫And so here, if I put a straight line here in blue, that's again, this interval, it's over. 59 00:06:42,960 --> 00:06:46,820 ‫And that's at the level of flyspeck. 60 00:06:47,890 --> 00:06:57,760 ‫Then this interval here, that would be the Delta Phi, the same like this one only now and do it here. 61 00:06:58,760 --> 00:07:04,220 ‫And this is again, it's over and just to clarify that. 62 00:07:05,360 --> 00:07:16,760 ‫So in that case, the time derivative of Phi Phi Dot K., which is this slope here at the point of Phi 63 00:07:16,760 --> 00:07:17,300 ‫K. 64 00:07:18,220 --> 00:07:25,370 ‫Since it's a slope is defined like this Delta Phi over T. 65 00:07:25,390 --> 00:07:28,780 ‫S over and like this. 66 00:07:29,910 --> 00:07:33,990 ‫Which is the same thing like Delta five over. 67 00:07:35,120 --> 00:07:49,930 ‫Delta T and so your fly sub, Kate, plus one equals five K, which is this one plus five, that's K 68 00:07:50,450 --> 00:07:52,070 ‫times T. 69 00:07:52,070 --> 00:07:58,130 ‫S over N equals five sub K plus. 70 00:07:58,130 --> 00:08:05,880 ‫And now instead of this slope, I'm going to write it down like this Delta five over T. 71 00:08:05,930 --> 00:08:09,320 ‫S over N times T. 72 00:08:09,320 --> 00:08:20,810 ‫S over N and of course you can cancel them out and you're left with five sub K plus Delta Phi and then 73 00:08:20,810 --> 00:08:26,660 ‫of course we'll give you five seven K plus one and this Delta Phi. 74 00:08:26,780 --> 00:08:28,130 ‫It's this interval here. 75 00:08:29,000 --> 00:08:40,490 ‫And note that this error here that you have this one in this version, it would manifest itself in this 76 00:08:40,490 --> 00:08:40,880 ‫way. 77 00:08:41,270 --> 00:08:47,540 ‫So you have a vertical error here, which is very small to draw right now. 78 00:08:47,540 --> 00:08:52,730 ‫But you can imagine that it's a vertical error, so. 79 00:08:53,740 --> 00:08:55,030 ‫Something like this. 80 00:08:56,190 --> 00:09:03,640 ‫So now you represent your states differently instead of seeing them as an area under the curve. 81 00:09:04,140 --> 00:09:07,770 ‫Now it's a no on the vertical axis. 82 00:09:09,250 --> 00:09:19,600 ‫But I wanted to show you this calculating area under the curve approach so that you would know why people 83 00:09:19,600 --> 00:09:23,320 ‫use the word integration when we compute new states. 84 00:09:24,630 --> 00:09:32,070 ‫OK, so that was the oilor method, and starting from the next video, we're going to learn something 85 00:09:32,070 --> 00:09:33,110 ‫more advanced. 86 00:09:34,320 --> 00:09:43,170 ‫We're going to learn an extremely popular integration technique in the engineering world called wrongly 87 00:09:43,170 --> 00:09:43,710 ‫Kutta. 88 00:09:44,730 --> 00:09:48,080 ‫So thank you very much and I'll see you in the next video.