‫Welcome back.
‫Now, we have been using a lot the word integration when we compute a new state.
‫So let's take a random state, let's say Phi Phi, Sub K plus one equals.
‫Five up subquery plus five Datsyuk times the sample time interval divided by RN, and we're saying that
‫we are integrating the state.
‫So why are we saying that?
‫Well, one way to think about it is that integration is finding an area under the curve.
‫So I have a graph here and here I have time.
‫And on this vertical axis I have five dots like this.
‫And then I have some kind of graph here for that.
‫So if the vertical axis is five dot, then.
‫Let's say that I have computed the area under the curve until this point here.
‫So what would be this white area?
‫What is its meaning?
‫Well, this white area is like is this and now let's take a new intervale.
‫Let's take this interval here in red, which is our PT's over again.
‫So this one this point here.
‫It's our FYE dot, OK, so it's this one here, that's this point.
‫So if you multiply five dot K times, T is over N.
‫Then this term here, it will be an approximated area under the curve, but only in this region in this
‫interval over and.
‫And of course, in reality, you want this interval to be small so that this error that you have over
‫here, this blue, this small region that this red area is not covering, so that it would be as small
‫as possible.
‫And so that means that your new state, fi, so K plus one equals your white area under the curve.
‫Plus, your red area under the curve, so you're fired.
‫Plus one would incorporate this and this.
‫And it makes sense, right, because you fight that Kay is your DFI deti, which essentially is your
‫delta fly over Delta T is just this is the differential case, the infinitesimally small case.
‫And this would be like an average slope and then it's over n well, it's a time interval, right.
‫So it's like Delta T. And so if you multiply this slope by Delta T, then they cancel out and then you
‫get your Delta Phi and so your fly.
‫So K plus one equals your fly up K plus Delta Phi.
‫So that's why it's an integration process.
‫Now that explanation actually differs from the previous course in this core series, but it's the same
‫thing.
‫Just explain differently.
‫In the previous course I said that you have time on the horizontal axis and then you have an actual
‫state on the vertical axis.
‫So now you have PHY on the vertical axis, not five dot, but actual FYE.
‫So now the area under the curve from the previous explanation is represented on the vertical axis.
‫So the number that you get here as an area under the curve is now here on the Axis five K, that would
‫be at this point here.
‫Because that's actually the main logic of integration in general, when you're computing the area under
‫the curve, when you take an integral of something, then what you're doing, you're creating some kind
‫of function where you keep your domain the same, which in this case is time.
‫But then the area under the curve, you mapped that on a vertical axis and it's no different here,
‫and if you find that sulphuric here was on a vertical axis, then in this version, it's a slope.
‫It's the slope of the graph.
‫So it's over, GNR was our time interval, so if you go from here up until you reach this Red Point,
‫which represents this slope, but the slope at this point here at K, not that K plus one, but at K.
‫So that's the slope here.
‫But then you projected one time interval ahead.
‫So once you reach this point here, this point on the straight red line.
‫Then that will be your face of Kate plus one, so this point, that means that their difference here
‫is Delta.
‫Why is this now?
‫And so here, if I put a straight line here in blue, that's again, this interval, it's over.
‫And that's at the level of flyspeck.
‫Then this interval here, that would be the Delta Phi, the same like this one only now and do it here.
‫And this is again, it's over and just to clarify that.
‫So in that case, the time derivative of Phi Phi Dot K., which is this slope here at the point of Phi
‫K.
‫Since it's a slope is defined like this Delta Phi over T.
‫S over and like this.
‫Which is the same thing like Delta five over.
‫Delta T and so your fly sub, Kate, plus one equals five K, which is this one plus five, that's K
‫times T.
‫S over N equals five sub K plus.
‫And now instead of this slope, I'm going to write it down like this Delta five over T.
‫S over N times T.
‫S over N and of course you can cancel them out and you're left with five sub K plus Delta Phi and then
‫of course we'll give you five seven K plus one and this Delta Phi.
‫It's this interval here.
‫And note that this error here that you have this one in this version, it would manifest itself in this
‫way.
‫So you have a vertical error here, which is very small to draw right now.
‫But you can imagine that it's a vertical error, so.
‫Something like this.
‫So now you represent your states differently instead of seeing them as an area under the curve.
‫Now it's a no on the vertical axis.
‫But I wanted to show you this calculating area under the curve approach so that you would know why people
‫use the word integration when we compute new states.
‫OK, so that was the oilor method, and starting from the next video, we're going to learn something
‫more advanced.
‫We're going to learn an extremely popular integration technique in the engineering world called wrongly
‫Kutta.
‫So thank you very much and I'll see you in the next video.