 it. Okay. So I guess this is project three, do some view plus maybe a more thorough example calculation. So I'll try to say kind of what what's happening that's new. So hopefully we remember we have sort of this analysis from from before. So we kind of chosen this this origin point to represent our leaf. We have some kind of normal vector we're associating to it. And maybe this is worth zooming in on a bit. Okay, we had some light ray striking at some angle. We changed the direction a little bit just so they're lined up sort of emanating from the same place. I think we called this thing the vector L. We had a few angles we were thinking about. One of them was this theta L. And maybe this time I'll emphasize that these are all functions of t. That just means that for L being a function of t, if I plug in a new t value, I get a new vector pointing in some different direction. That's what it means for a vector to depend on time. And for the angle depending on t, this is kind of more like what we're used to, I plug in a different time and I get a different angle. But just remember from sort of previous discussions that an angle isn't really enough to determine the vector, it's just kind of one piece of it. We have an angle. Let me also have a magnitude we have to worry about. And so I'm just going to shrink those a bit. So we have some room to work with if I can. Okay, so there's that angle. We had this angle here, which I think I called theta Ln, which also depended on t. And this was kind of the one we were mostly concerned about, but kind of an intermediate step was finding theta L of t. Or rather, we're kind of choosing some function to model that. And maybe what I'll mention here is that that this thing was a right angle before, maybe I'll just call this alpha. And this was set at pi halves. So this is supposed to be where the leaf is. And this is like a plane of the earth. Yes, we can do that. Earth is a sphere, right? But if you zoom in really, really, really close to something, it starts to look really flat. So we're going to do this like plane approximation, if you're the size of a leaf compared to the size of a planet. Okay, so we have this kind of situation is what we had before. And let me actually add something here to emphasize that maybe this was part of the data of the leaf, everything in green. The leaf kind of sat at some angle, which we were assuming to sort of be essentially zero, right? It was just lined up with the ground. And we had this normal vector pointing out of it, just directly out. So it was at an angle of pi halves. So the situation we want to think about now is essentially, so this is sort of the first part of the analysis. And we want to modify this just a tiny bit. And let me see what can we keep? Actually, maybe we have to delete pretty much everything. Can we do this? The idea is now, we still have this sort of situation where the leaf has a set origin. But now maybe the leaf is pointed at some initial direction. So maybe something like this. And okay, we still have some, maybe I'll draw the light ray in a slightly different position here. Okay, so essentially the same situation as before, I've just taken the leaf and I've kind of tilted it a little bit. And the thing that's changed is that now we have, what did I call this before, alpha, maybe put three, three angle markers on there to denote that these are supposed to be kind of measuring the same thing. And as before, we have this one was two, theta ln of t and theta l of t, something like that. So just remember theta l of t was really, it really doesn't depend on the leaf at all. So it's really just measuring kind of where the sun is in the sky over time. And theta ln of t is measuring the relationship between the leaf and the position of the sun. What alpha is, is alpha is like what is the initial angle of the leaf? Just as you find it, it might not be flat, it might be tilted as Q. In fact, it probably will be. And I think for part of this project, you actually need to like go out and find a leaf on a tree and kind of estimate this angle. So you might have an angle like this pointing off, you know, to something off to the left hand side, you might have something where it's kind of rotated the other way, where the angle is bigger than pi halves. So this is kind of the new thing. And so we need to do a little bit of analysis here to figure out how, I mean, how does this play into things? So the first thing to notice here is that if I do this, I can take alpha plus, sorry, I guess I should mention that this angle here is still a right angle. It's just that, yeah, the normal vector coming out of the leaf will always make a right angle with the leaf. But we're worried about the angle that the leaf makes with the actual ground. So that's what this alpha is. But if you just take alpha plus theta ln of t plus theta l of t. So alpha doesn't actually depend on time, it's just a constant. These other two angles do depend on time. But they have to satisfy a relationship at every point in time. And that's namely that all of these, when you add them up, just have to be equal to pi. Because if you imagine at some early time in the day, you have a very small theta l and theta ln is very large. And it fills in the gap between those. If you just add alpha under that, you get the full sweeping out of half circle. But just kind of moving the light vector around just changes the relationship between theta l and theta ln. And when you add alpha to it, you still end up getting, you know, the full half rotation. And so what's nice about this is that if you have, so maybe you have functions for these, okay, maybe you have a function for this one, for example, that you've come up with and you've sort of justified why it should make sense in this context. Then you can just write theta ln of t is equal to, let's see, it's really, we just want to move everything to the other side. So pi minus alpha minus theta l of t. And that's probably a good formula to have on hand. Namely that you really only need to do theta l of t and then theta ln of t is related to it in a certain way. So these functions aren't really independent. Really one of them depends on the other one. Okay, so I mean, then the task is, okay, what is, what is this thing? What is theta l of t? Or what's a good choice here? So if we're trying to find out kind of what function this should be, I would do something like this. I would kind of go to my picture and I would think about some different values I can pick out. So I know that maybe theta l at time zero. All right, this is just the plane, or sorry, the angle that the light makes with the plane of the plane of the earth. And so maybe I know that theta l of zero is just equal to zero. I'm just trying to find some interesting points on this graph. I know that theta l at 12 noon should be pointing straight up straight up. So theta l at 12 will be pi halves. And maybe theta l of 24 should be equal to pi. All right, this is the situation when the orange line is pointing straight to the left. And if I just kind of assemble, and I kind of know, plus it needs to be periodic. So maybe theta of l 20, or sorry, maybe t equals theta l of 24 plus t. Just means that if I go out and measure the angle at 3am today, and then I go out and measure it again tomorrow at 3am, well, I'd really hope to get the same angle at the same time, even if it's a different day. So this is what we mean by periodicity. And okay, if I have this, now I can start to assemble some kind of graph of what I think this should look like. So what do I need to measure on the axes? I need a theta l as my dependent variable, and it depends on t. And I have some points. I have zero zero is a point. This is 12. And pi halves is a point. And pi 24 is a point. Okay, so there's sort of a lot of functions that could do this. You know, maybe does something like that, you know, saying you can sort of come up with any function you want here. What I'll do for the example is we'll just kind of go with what's in the project handout and say, let's assume it's linear. So I kind of missed the point, but let's make it bigger so the line actually goes through them. Okay, let's label these 12 by halves, and 24 pi. Okay, so now we've, we check this off this off and this off, we just need this periodicity thing. So maybe what I'll do is assume that it just repeats. I'll assume that like 24 hours, we have one full period. And somehow this is the whole thing with periodic functions is that we only ever need to know one full period, because we know we can just kind of duplicate and mirror it everywhere else. So we know that it kind of continues like this, or maybe I'll just even draw it in with an actual line there, but it's important to know that this is just one full period. Okay, so this is some function theta L of t, giving this line. And how do we find what the actual function is? So f of t, it's a question mark. Well, I guess I can just do point slope, right? So let me just pick, okay, let's just pick two very easy points. Let's do, so use point slope from our distant, distant memories in this class. I have two points. So I can determine the equation of a line. So I don't quite remember what the formula is, but I know I can calculate the slope. It's always like a change in y over a change in x. But look, one of the points is zero zero. So the change in y is just pi halves, change in x is just 12. So this is pi over 24. All right, we're just, you know, we're subtracting a zero off in the numerator and the denominator. So we just don't even have to worry about it. And just double check to make sure this matches up. Yep. Okay. So we have a slope. And we can just patently see what the intercept is. I think here we want b equal to zero. So maybe what we actually want for this function is it to be like a closed circle and zero, and maybe an open circle at 24. And then a closed circle down there, continuing in that way. Right, because we do have to make sure this is a function. If we filled in both of those circles, I would be looking over one x value, and I would be seeing two different y values. So what is the function return? Well, it has a choice. Functions don't know how to make choices necessarily. So we have to pick one for it. But okay, we have a slope, we have a point. And that's all there is to it. Theta L of t is pi 24 t and plus zero if you want. But let's me have we have some function that's fit to that. And importantly, this is in radians. It's like I kind of mentioned in a previous class, like you really just want to be working in default in radians whenever possible. It kind of makes makes life a little bit simpler when you're doing these equations. Otherwise, you have to put in some conversion factor involving a two pi and a 360. So you can do that if you want, but you have to pick one convention and definitely stick with it for the whole project. And it's really nice about this is let's see we had an equation somewhere. Here we go. This one for what theta ln was just drop it here. And now we know that this is pi minus alpha minus pi 24. Okay, so we have a model for this function too. And let me, yes, I might have a demo that might help explain what this should look like. One sec here. Okay, yeah, let me turn off some stuff here. Okay, so here we just have theta L of t just at zero zero and 12. Well, it's going to give us some numerical approximation probably, but you have 1.5 is approximately pi abs. That seems to work and 24 is pi up here 3.14 ish. Okay, then if we just do theta ln, that's the function we get for it, which seems to make sense. Right, so it should be decreasing throughout the day. It should start off at a max, right? If you have the angle of the leaf being orthogonal to the angle of the sun at time zero, and as the sun is sort of rising, then the angle should be getting smaller. But yeah, then there's there are a few things to consider here. One of them is that eventually if we go back to this picture, this one here, one of them is that if your angle is kind of pointed off to the left like this, you'll have a small issue because once you let's see if you're like here, for example, then it'll actually be so you'll have like sunlight hitting the back of the leaf instead of the front. So it's not going to be absorbing any actual energy until like some fixed time, like whenever it passes this plane. And you'd have kind of a similar problem if the angle was kind of pointed off to the right hand side, like eventually you would be going over here and like you would be passing the plane on the left hand side. And the sun would be coming in behind the leaf. And so you should be zeroing it out for that too. So all I mean by that is that at some point, so here I'm just modifying like how far this function goes. At some point it actually needs to be zeroed out. So there's actually a little bit more to do here than just kind of like writing down this function this way. You're going to have again a periodic function, but it's going to kind of max out at a different place and it's going to be periodic with a different period. And so you might have something that ends up more like a tent shaped once you kind of figure out how to right. So if you want to go back to the physical situation here, we know that like at least once you're past this point, you're kind of in the the original case where the leaf was sort of flat. It's just that we have to zero it out sort of outside of those cases. And so you should have an angle starting and something like zero or something like my halves. It should be decreasing to zero. And then maybe it should be increasing again. But so you'll you'll need to kind of do some sort of mirroring because whatever's happening and happening on this right hand side of it should essentially do the same thing. It's happening on the left hand side. So what'll happen in the graphs here is that now you have some sort of shape of a function to start with. And if you're going to actually come up with what your final function will be, this will be like theta l n of t, so it'd be t. Right. So we had this, this was kind of our original function was just something linearly increasing. We did this thing, we got something that was decreasing but what'll happen is that we need to maybe like mirror this somewhere a little bit and maybe like zero part of this out. See here. Right. So you might have something that's more like this. And then maybe it's zeroed out from some specific time. So maybe this is I don't know, 8, 8 p.m. or something. This is zero. This is kind of matching up with the situation we had originally like, or wait, no, so let's see in this situation we had zero up until some time. So let me switch this up a little bit. So we started off with something kind of like this. Just erase a bunch of this. Okay. Yeah. See it's something like this except for we have to kind of zero it out for maybe a first part of the day and then maybe it'll do something like that and then maybe we'll have to kind of reflect it up like that to denote. So this is sort of capturing the fact that sort of back up in this picture again at some point you're like directly lined up with the vector and then the light vector starts or the angle starts increasing again. So you go somehow up to to the angle is decreasing down until it's zero at some point in the day and then it starts increasing again once you once you have the vectors lined up. But so you should have essentially like a line with the same slope or something if you're doing something linear. You just kind of reflect it like that. And so you get something maybe a little bit more weird for your periodic theta ln and then maybe this is 24 or something that's zero. Maybe this is happening at 12. So it won't be entirely symmetric. You'll have this kind of like check mark shape maybe then you'll have to decide here too. I guess you want to fill in this one and maybe leave that one out so you get an actual function out of it. If you don't get something exactly like this it's totally fine. If you just come up with a different theta ln to begin with that's totally fine too. But as long as you're somehow incorporating this analysis and saying how it should match up with what's happening in the picture then you'll be fine. Yeah so how do you actually how do you find this point? Essentially you want to do some kind of analysis like this. Here is the original situation. Something kind of like that. That's the normal vector. Maybe that's our light vector. This one was alpha. This was theta ln. I'll just drop the t's to keep things quick for now. This was I'll just make it all the way out here. This was theta l. So you really want to find what this angle is here. Let's call it psi. So what's happening is that you have no energy for I guess theta l maybe less than psi. And okay how can we actually find that angle? I'll have to use a few things. This was orthogonal. We can extend this. We can use this thing from geometry right. If I have yeah so two skew lines then like the opposite angles are equal to each other. So like this would be psi. And then this again would be a right angle. So we can use the fact that alpha plus psi equals pi halves. And okay but alpha is something you know so you can just solve for psi. So you get some angle and then you just need to solve for t and theta l of t equals psi. And this will give you some like t zero or something to work with. And if your angle is pointed the other way you just have to do sort of a similar analysis where this thing is mirrored around. I'll just kind of sketch the picture really quick. Maybe your leaf is pointed this way. Kind of doesn't really matter where you point the light factor. As long as you're consistent about it. And in this case let's see this is now the angle alpha you want to measure. And this is kind of the so here here was kind of the the problematic like sector when we were doing it before. There's just no energy in this kind of red zone. And here's kind of the problematic sector if it's pointing that way. There's going to be no energy if you're in this zone over here. So you just have to do some kind of similar analysis but it should go pretty much the same way. If you didn't catch all of that don't worry I'll post up notes for this. Just make sure I save this. There we go. Okay so hopefully that handles the case where you have your leaf pointed at some angle. So I want to say a little bit about how to fit an energy density function. And before I do that maybe I'll just maybe we can take a quick one minute break. It's a lot of mathematics all at once. So I'm going to grab some coffee. If you want to get up and like stretch your legs or something feel free. We'll start back at like 48, 47. Okay so hopefully people are ready. Yeah so this energy density one thing I found that was like maybe a little bit confusing is that this whole analysis is essentially independent of um the leaf itself. This really just has to do with like power output of the sun and sort of relation of the sort of how the earth sits in space with relation to the sun. You can imagine like if you're at a point on a sphere or something like that. Let me draw this a little bit easier. And so we're at like the north pole or something. If the sun is over here these light rays are coming in parallel anyways. So there's just kind of no way you're going to be absorbing any light. And then you know maybe after I guess really we should be should be rotating this right. If we do this kind of orbits around a little bit. Okay well this is this is a terrible non scientific drawing so please bear with me but you know if this is something more like noon then you have light rays kind of directly hitting that point. And then this is kind of at you know maybe the end of the day more like sunset. And you have something like that. I have no idea if this is the actual way that orbit works but this is just a schematic that hopefully helps you reason about why should this change over time. Will the earth and the sun or you know locked into some orbit together. And you know the amount of energy density is going to depend on how parallel these rays are hitting some small point on a sphere. So all of these things. So just in general we know we want something that sort of looks like this. So if I just kind of start with a graph and I make some awful scatter schematic about what I want hopefully I can reason from it to come up with some actual formula. So I know I want this to start at zero zero. Well okay so there's a caveat here. Let me just make a note over here and I will say what this is after we've looked at some more of this function. This is mostly just a reminder to me. But okay so maybe zero zero is a point we have maybe 12 and I think it's like point 0.136 is a point we want. So maybe there and here's 24 zero and okay so these are some points we want and then we also want it to be periodic. So we're just kind of considering one period of it and we want to extend this out to all of the real numbers by just picking it up moving it over one period, dropping it down and just duplicating it that way. Okay so we expect that this is not linear like before and so one thing that you could do is just attend and you could find lines for that and you would get some maybe like piecewise equation for the two lines. But we expect that this varies more smoothly than the other one. Well it's not a constant rate of change like somehow you know the change of the angle in the sun over the leaf is like a constant rate. But the change of this energy density is like you know somehow the rate of change itself changes which is like a very very difficult concept to look at until you've taken calculus but you kind of want something with more curvature to it is the best way to say it. So maybe something like this, this is not going to be the exact function we get but you know we want something that's maybe more smooth like this. And so one idea is to try to fit a fit a wave. So we're going to assume that this has kind of wave-like behavior and maybe what I want to do here's kind of a microcosm of what I'm thinking is maybe you have something like this and then just have it repeat kind of like that where this is just like some chunk of a sine wave or a cosine wave or something that I've cut off. Okay so I need to try to fit some kind of wave to this so maybe I will think of this as a sine wave and okay so why did I choose sine? Well because sine is something right if you remember kind of what sine looks like terrible sketch something like that right so it starts at zero zero so that's a nice property of sine versus cosine remember cosine starts up here so this maybe starts to look like what I want and then what I would have to do is realize that this would be like essentially just this part would be my entire graph over here so from all this conclude maybe e of t is maybe half period of a sine wave my half period I mean I have this kind of like parent function here this is be a little bit careful here this is sine of t and I'm just using a half period of it to model my my e of t okay so this says that we just try to run through this derivation I can do this kind of general form so I need an a sine of omega t minus so usually label this phi plus delta or something okay so this is like I put in all of the parameters I possibly could to control what this this wave does and this is where if I were you I would start doing some like exploratory analysis on like in Desmos or something so let me pull this in like if I were trying to do this project myself this is exactly how I would do it is before I even try to solve this analytically and find out what all those parameters are I would like to know how do these parameters even affect what we're talking about so this you know I've just put in some function where I've made all of those things parameters um f t and okay amplitude if I just kind of animate it out I see kind of what it does I should be careful I guess between the amplitude being positive and negative but I'm seeing that all it does is change the the sort of the displacement and sorry I guess I should change this to zero to get the the usual one but all it does is change the the sort of uh where the peaks are landing I have to be a little bit careful because if there's some displacement like this it's not that you know here like that we're seeing it's peeking out at like eight but the amplitude is four so really the amplitude is like displacement from some like dc level of the dc being like direct current or some like ground level or something if you think if this is like an electrical uh wave um if I do y equals d okay not quite I called it s here there we go yeah so it's really that this like horizontal line is some like ground current in the amplitude is measuring like how much do you wiggle around that so I think for us we'll probably be safe enough setting s to zero because zero zero was a point we were interested in so that seems fine amplitude will probably be determined by um that maximum we wanted sorry we're only going to do one half period of this thing and I guess I have a cosine here but I really want any sine oh but I guess that was a point I meant to make is that you could do all of this with cosine and sines if you wanted to here's down here is a function with sine um and the only difference between the two is that if I plug in well okay let me just do this if I make these exactly the same the only thing that's different between them so let me just set these back to like the that should do it yeah so the only difference between these is that um sine and cosine are just phase shifted versions of one another so sine wave and a cosine wave are basically the same thing it's just that the I mean depends on how you want to look at it maybe the sine wave lags behind the cosine wave a little bit so if I just put in a correction factor of minus pi halves or maybe it's plus then they exactly line up okay okay so that's all there really is to that so I could pick a cosine wave and then just like shift it or something like that um but yeah so maybe just kind of looking at a cosine is well let's just do a sine okay so s is going to s is right the thing I'm adding on at the end is control how high up it goes p inside is this like phase shift and what happens here is as I move the phase all it does it's just going to pick up the sine wave and literally just shift it off to the left or right this is exactly the same thing as the the translations we did earlier in the class you know a minus p of a phase shift is going to shift it to the right by p units the only thing that's tricky here is that your units might be might be whole numbers or maybe they'll have to be like multiples of pi or fractions involving pi or something okay so that's all a phase shift does just kind of move it left and right and then just the frequency I mean as we run the frequency up it's just telling you essentially how many um how many periods occur in say like one unit time or in this case in two pi units of time um and you can sort of normalize this to get how many how many periods occur in exactly one unit of time if you want there's a formula for that but you won't need it here it's just an idea of like what are these parameters actually control so it seems like a is the amplitude this this should correspond to that maximum energy we were seeing the frequency somehow going to be tied to the length of our period of 24 hours so we'll need to kind of make that work and this phase shift probably not so important for us because we're already we're starting at something like zero zero and if s is zero I think we're good okay so you have to do a little bit of some kind of analysis to justify where all of this is coming from but you know you can kind of read off from this that the amplitude should be equal to 0.136 so essentially we're taking this graph here this is t this is our e of t and we're trying to first without worrying about these two just kind of fit a cosine wave here or sine wave I guess this is the one we're kind of looking at and then what we'll do is we'll chop off this part so we need a sine wave that fits this where 24 zero is where the half period occurs as opposed to down here this would be in just a normal sine wave I guess this would be pi zero but okay so the amplitude is just how far it's deviating from zero that'll definitely be just this max value we're picking up here let's see phase half i be equal to zero because fortunately like it already horizontally lines up with what we want we're hitting zero zero with our sine wave delta be equal to zero two because we don't need to shift it up or down so really the only thing we have to worry about is well I can start with this I know that the period of this sine wave okay so this is a little bit tricky the period actually needs to be 48 hours for this sine wave and why is that thing is that we won't we only want to keep this like this first half of the sine wave and so we need to like have our model function be periodic with period 48 because that's what a usual sine wave does right it has sort of this part where it's above zero and then has this other part where it's below zero we're going to chop off the part where it's below zero later but before we do that we still have to just build the sine wave to start with in this sine wave it has a half period of 24 so it has a full period of 48 but okay we know that that's 48 and we can also use this to relate it to the frequency we know that I guess this is coming from the formula t the period is equal to two pi over omega so omega is probably the frequency and so we get omega equals hopefully I got this right pi over 24 okay so let's assemble all of this information now so we get something that looks like e of t is equal to 0.136 times sine of pi over 24 times t I think that is it for this part and I would actually go and try to to graph this before you actually use it um okay say zero point whatever it was 186 and omega was something like two pi or sorry pi over 24 okay so it looks flat that's maybe a good sign change our window a bit so x's should be like t's here zero to 24 and then y should be like zero two okay so that's starting to look better now that things are on the same scale and I need to set p equal to zero s equal to zero and then this needs to be a sign instead of a cosine okay so she gets something like this it should hit the right points zero zero 24 zero and whatever this max point is 12 and zero point 186 and all right that's the exact kind of thing we were looking for and now you just need to say how to you really have to be a little bit careful with all of these say on zero to 24 for example or maybe we leave out the endpoint and you want to say something about how to extend this periodically okay so in the last 10 minutes here just say something about the total energy so you can kind of see what expression you're supposed to get um so I think it ends up being kind of complicated um it's good to know like what level of complication to expect okay so maybe before I do that were there any questions on anything up to the energy part yet I think this total energy we were calling gamma of t and from one of the last classes we found that this was essentially the vertical or the y component of that um l vector of t times some area okay and this thing you actually have to estimate so like go out and find a leaf and I think this is centimeter squared what the units are supposed to be there and I think we found out that the derivation here led us to expand this part out as e of t that was something like sine of maybe theta l of t or theta l n of t so you'll want to run through that derivation yourself to make sure it makes sense and okay now we can just start really plugging things in um this is 0.136 so e of t is the thing that we we just got up here so it's all of this stuff and now I need to take this stuff here and I have some formula for theta l n of t as well I think it's something like this and times a okay so that's what maybe this energy function will look like after you've kind of plugged everything in assembled it all together um yeah I will say here that there is some way to like if you want to simplify these two signs there's a way to combine these but you don't really need to just having some formula for it is well and good so now you want to plot t versus z and I forgot I totally forgot the caveat that I meant to mention up here so the caveat is that you may need to shrink this and what I mean by that is that you might have to redo this analysis with some additional assumptions like maybe the sun is not up like we know it's not up at all not hitting the earth or hitting this point on the earth maybe say before 6 a.m or past 6 p.m but I claim that essentially everything we've done here goes through exactly the same way the only thing we're changing are these two little anchor points down here two six zero eighteen zero and so what you'll get is here we got some you know part of a cosine wave like this the only difference is that your cosine wave will be fit in like that and then you just have to zero it out for this you know up until 24 or something like that and extend that periodically so that's the caveat is that you might have to do a little bit of extra or do a little little bit of different work if you're making that assumption okay so maybe what happens is that it's you do this uh sorry this should be a gamma use this gamma of t maybe you like try to graph it or something and you find that it's just zero in some region and then it should sort of grow and fall back down like that okay so like some of your graph might be down here on the axis for this part down here on the axis for this part this is 24 and so what you'll want to do is figure out where this there's just like t zero here and t final here and you chunk this up into a bunch of t's so there's t1 t2 t3 and t4 and we're going to approximate the area under this graph and I want to see if should I say units here uh yeah see that in one second here so the way that we're going to get this area is that we're going to approximate this by rectangles essentially uh using their left end points so kind of miss some here we'll pick up a rectangle here something like that I should do these in a different color sorry okay so that's one rectangle we're just taking the left end points of all of these t's trying a rectangle to the the next t okay so this is all the area we're picking up in the approximation I'll draw these in a slightly different color we're picking up some extraneous area and some well so we're missing some area right so we're not hitting these parts and we're picking up some extra area that we don't want uh you know sort of in these spots so what'll happen here is that the area of this it's like the total energy it's going to be equal to the total area here and it's going to be the area of the rectangles plus some error that depends on n and so this is where the red stuff is going into the air the blue stuff is going into the air the green stuff is going into the the thing we compute and so just so you have an idea of of what's happening here uh sorry I realized we're basically out of time let's do one last demo here so you can kind of see what's up yeah so this is kind of what's happening here where we're taking some boxes to approximate the area it's not exactly right the idea is that if you take so in is controlling the number of boxes here if we were to somehow take a bunch of boxes and it gets way closer to the area and kind of in the limit if you just take infinitely many whatever that means then you would basically get the exact area or this error would go to zero so that's kind of what's happening here is that we're taking a few boxes to make an approximation and this approximation could get better if we took more boxes okay maybe I'll just list out the formula that you might need here sorry just one sec my notes closed up so what you will get here is let's just call this thing b1 for this box b2 b3 b4 you just compute the area of say b1 okay what's the height of this rectangle well it's gamma evaluated at t1 just literally this point here is t1 gamma t1 so the height of this rectangle is the y-coordinate and what's the width well then it's t2 minus t1 t1 so I'm just a delta t and so you're just going to add that to area of b2 equals some same thing except you're going to plug in gamma t2 now it'll be t3 minus t2 so on and so forth so take a look at the video that's posted in elc cal shows a nice way to like do this in a spreadsheet so you don't have to compute I don't know a million of these and then just be a little bit careful because the gammas need to eat in hours but I think part of the project asks you to break these into maybe like 15 minute intervals or something like that so you may have to do some some conversion somewhere so like if your t0 is zero and your t1 is 15 minutes you might have to do you know that's maybe one quarter of an hour or something like that change it to either all hours or go back into your function for gamma and put in a conversion that changes hours to minutes there but I think that'll be way more complicated because gamma is a huge function involving t's everywhere so it may just be easier to multiply this by so this thing should be watts per second this thing will be hours so you just need to multiply by seconds per hour to get the right units for that it's like 3600 or something okay um so I'm sorry kept you guys way over I'll let you guys go I'll also stick around for like five minutes here if anybody has any other questions and I'll try to schedule an office hour today and announce it on ELC do you guys have any any questions you want to ask or glad I'm glad it helps and it's a it's a pretty complicated project but all right so I'll go ahead and in the meeting just feel free to email me if you need anything