 So today what we're going to do is we are going to apply a very common numerical method to calculate for calculating the area of irregular shapes. And in this case we're going to apply it to the plan form of wings. Now we're going to keep our wing shapes relatively simple today so we can demonstrate the methods more than we actually worry about the details of the mathematics. So here's how we're going to start. First of all we're going to look at the lower wing in this case and you'll notice the shape of this wing is not particularly complex. And one of the things I might do in this wing if I were to calculate the area of the wing is I might start by taking the wing and dividing it into two parts since it's symmetric. So I'll divide it right here down the middle and I'll recognize that the wing is effectively now that I split it into two parts in two shapes. And these shapes, each of these shapes, they're the same, okay? But each of those shapes is what we call a trapezoid. Now we have a formula for the area of a trapezoid. The area of a trapezoid is very much like the area of a triangle. The area of a trapezoid follows a rule that looks something like this. The area is going to be equal to one-half a base, something known as a base, times the average of two heights. Now we have to identify what we consider to be the base and what we consider to be the heights in this particular case, okay? Well, in this case we have some terminology that we use for wing plan forms, it's a little bit different. We have our span and we have our cords, where our cords are from the leading edge to the trailing edge. So if I rewrite that in this case, let me go ahead and identify a couple of pieces here. First of all, we have a measurement here, which is the cord at the root. So there's our root cord and that's a measurement we can make in just a moment. And then we go out to the tip of the wing, either side. And we have a measurement there, which we can make called the tip cord. And then we can also make a measurement between those two things along the length of the wing, which is called the span. Now if I want to find the area of these things, I can look and see, I see, okay, I have these two cords, those are going to fit in as this H1 and H2, the heights at different places. And we can consider the span to be the base. And notice we're going to, we can calculate that for one side here, one of the sides of the wing. And then we can just double it for the other side. So I'm going to go ahead and do that calculation here, just to be clear on how we would go about doing that. And first of all, I got to look at my system and say, okay, what units am I using here? And I'm going to decide that my unit is going to go ahead and be one of these little segments here. Okay, one of the segments between the red and the blue, instead of making that half a segment. I could make it a whole segment or half one, but we're going to call this one unit of length. All right, well, if I look at that as one unit of length, then I can go ahead and say, okay, so what is my span? My span appears to be 1, 2, 3, 4, 5, 6, and 8, 9, 10, 11, 12, 13. We'll say about 13, maybe 13.3. So my span is 13.3 units. And then I will look at my tip cord, which looks to be about 1, 2, maybe about 2.6 or so, but then there's a little bit less there. So we're going to call that, we're going to estimate that as 2.4 units for the tip cord. And finally, over here for the root cord, starting up here about 1, 2, 3, 4. And we'll say that that's about just under 4.5. We'll say this is about 4.4 units. And so there's my measurements. And I could put those measurements in to find the area, which I will do now. So my area is going to be equal to 1 half times that B, in this case, that's the span, 13.3 units, times the sum of my two cords, 4.4 units plus 2.4 units. And when I do the math there, I'm going to get, let's see here, 4.4 and 2.4 actually the average out to be 3.4. And then 3.4 times 13.3 gives me 45.22 square units. But let's see here, I can only keep, it looks like at best I can keep maybe two sig figs there. But let me go ahead and write that that's not 44, it's 45. We'll write 45.2 units squared. And then we realize that that's one side, so we multiply that by 2 when we find the area of this wing to be 90.4 square units. Where the best I could keep is 3 significant figures, I'm probably actually better suited to saying only 2, so maybe I should say it's about 90 units squared. So hopefully you can see the process whereby we calculated the area of that wing. However, now we're going to look at the wing that's up above it. Now if you look at that wing, you can say, oh, that's an ellipse. I happen to know the formula for an ellipse, so we might be able to calculate that using the formula for an ellipse. But that's not what I'm going to try to do right now. What I'm going to do instead is I'm going to recognize this method that we just used where I took the wing and I sliced it vertically and created trapezoids. And if I look at this wing, I'm going to do a similar thing. I'm going to take the wing and I'm going to slice it vertically. And when I do so, I'm going to make a slice there, I'm going to make a slice on every one of these red lines. I could slice it as much as I would like, but I'm going to go ahead and slice it on each one of those red lines. And when I do so, that's going to create a bunch of little segments. And I'm going to go ahead and label each of those little distances there. Okay, so I did all those slices. And when I've done all those slices, I'm going to go ahead and give them labels. In this case, I could call them Hs, but I'm going to go ahead and call them chords because there's a series of chords here. So here is my chord and I'm going to call this chord zero. And this is chord one and chord two, et cetera, chord four, chord five, chord six, chord seven, eight. All right, so there's 13 vertical divisions, marks here. Notice, why did I start at zero? Well, I started at zero because when you make slices in something where you have a start and an end, then if you start counting the number of actual pieces there are, if I look, there are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 pieces. So to keep the number of divisions and the number of pieces somewhat comparable, we have 13 divisions and 13 slices plus the zero slice. So it's usually a good habit to start at zero when you're doing something like slicing, okay? We're also going to look at each of these. There's also a measurement that's involved here where each of these slices, I did a very specific choice. I made a choice to do a piece of the span. In fact, let me go ahead and make that in red because we were making the span in red. So there's a piece of the span and I'm going to call that delta S. It's a change in the span. And notice that is 2.0 of my units. And I was consistent about that, that they are evenly divided across the entire span of this wing form. Now you'll notice I wasn't perfect. There's a little bit of edge that I cut off over here and a little bit of edge that I cut off over on that end, okay? That was a decision that I made that it didn't quite fit perfectly. So I have made an approximation. If I calculate using this, I've chopped off those tips a little bit. If I think that's really important, maybe I should do a better job with my division making sure my lines fit a little bit better into the shape. But I'm going to make another approximation here. I'm going to look at each of these shapes and I'm going to realize that even though this is a curve, because I've sliced it up into enough slices, that if I look at any one part of this curve, like it's segment right here, this is section nine. If I look at section nine here and I draw a line between those two sides, I'm going to create a shape. It's going to be a trapezoid. And that trapezoid is going to be an approximation for that curved shape that is actually there. I drew a straight line and that straight line doesn't quite perfectly fit those curves, but it's a pretty good approximation to those curves. And you'll notice that if I slice this up more carefully, the smaller my slices, the more likely that line is going to be close to that curve. So there's kind of a rule here when we're doing numerical work. More accuracy equals more work. I could get more accurate by slicing up more lines or by taking them and being much more careful about making sure there's nothing hanging out on the edges. That's going to take a little bit more work and I have to decide is the work worth the accuracy or how accurate can I get. In this case, I've decided to slice it up every two steps. In the case below, I sliced it up every 13.3 steps, which took me a lot less work. And we were pretty lucky because it's pretty straight. It looks like those lines are straight, so it wasn't that big of an approximation. But if we had tried to do that here, that curve would have made a very big difference. If I tried to make a line from this middle down to that point, you'd see I wouldn't be doing a very good job approximating that curve. So making more slices means I'm doing a better job. Okay, well if I recognize now that I have a trapezoid, and I see here that I have a trapezoid, what is the area of that trapezoid? Well, each trapezoid in this particular case has an area that follows that same formula, one-half base times height one plus height two. They have that same area, but let's go ahead and I'm going to use this space here to sort of write this again. Okay, so our area of one piece of the trapezoid is going to be equal to one-half. Whatever that delta S is, there's our two units there, that's the thickness or the base, and then it's going to be equal to our chords. In this case, chord 8 plus chord 9 for this section number 1, 2, 3, 4, 5, 6, 7, 8, this is area A9. So area A9 would be one-half delta S, C8 plus C9. Notice I could do that for all the areas. I could say, okay, what about area 10? And I could say area 10 is one-half delta S, C9 plus C10. And repeat the process for area 11. Area 11 equals one-half delta S, C10 plus C11. So hopefully you can see the process that's going on here. Okay, now I started at A9. I should probably start back here over at A1, where A1 is delta S plus C0 plus C1. And notice as I do that, once I'm done, I'm going to take all of these areas and add all of them together to get the total area. Okay, that seems like a reasonable process. Hopefully you can understand what I'm planning on doing there. So let me go ahead and take a look from that. Let me erase this section down here and rewrite that a little bit and think about the process of what I'm doing. So I'm starting to find my total area. I'm going to take area 1 plus area 2 plus area 3 plus all the areas in the middle plus area 12 plus area 13. And when I look at each of those areas, I'm going to look and I'm going to see, okay, area 1 is one-half times that piece of the span times C0 plus C1 plus one-half times the piece of the span plus C1 plus C2 plus one-half the piece of the span C2 plus C3. And we're going to keep going with that process. Well now you'll notice, hopefully, that all of these terms have some things in common. They all have a one-half and they all have a delta S. There's the one-half delta S and the one-half delta S. So doing a little algebra, I can say, okay, well my area must be equal to one-half times delta S times all of the pieces that are in the parentheses. And let's start adding them together. There's a C0 plus a C1 plus a C1 plus a C2 plus a C2 plus a C3 plus a C3 plus a C4 plus and we keep going until we get to the very end when we'll have a plus a C12 plus a C13. Hopefully, you can see the pattern there. Now, one thing I recognize, the last piece I recognize here is there's some repetition in here. You see the C1s? There's two of them and there's two C2s and there's two C3s and two C4s and two C5s. In fact, there are always two for all of these things in the middle. They count twice because they count once for the one that's to their left and once for the one that's to their right. All of them get counted twice except for the ones at the end, the C0 and the C13. And this pattern leads to something we call the trapezoidal rule, trapezoidal rule, that if I take my area, I can find my area by dividing something up into little segments. I have this factor of a half because it's a trapezoid and that's the average of those two sides. I have this thickness that I divided them into, chopping the span up into little pieces. Delta S is the change in those positions of those little pieces. And then I'm going to put in one times the one at the beginning and a one times the one on the end, number 13. And then all the rest of them get multiplied by two. Two C1 plus two C2 plus, et cetera, plus two C12 plus one C13. And this is known as the trapezoidal rule. And we'll apply that trapezoidal rule to find the area of complex shapes. Now there's another rule, remember I said more accuracy equals more work. There is another rule that we can use. This rule actually takes a little bit more work, but it is a little bit more accurate. This rule is known as Simpson's rule. And in the case of Simpson's rule, we use this same basic idea of slicing things up and dividing them. However, as it turns out, we have to be very particular about our choice of slices. And let me show you why. We'll start off with Simpson's rule. We'll say, okay, our area is equal to, well, our factor out front in the trapezoidal rule, that factor is a factor of one-half. In Simpson's rule, we're going to count things more, and so that factor actually ends up being one-third. We use the same width that we use in the trapezoidal rule. In this case, our delta S we would use in this particular case. And then the pattern looks something like this. The very first unit, 1C0, the very first chord that we would use would be of a factor of one. The second one, however, has a factor of four. The third one has a factor of two. The fourth one has a factor of four. And then we go back to the two again. So the pattern is one, four, two, four, two, et cetera. And if we look at our example above, we start here, it would go one, four, two, four, two, four, two, four, two, four, two. And then we would go back to one. However, you'll notice a one is next to the four there, and a two is next to the four there. So that is actually incorrect. Our example, we can't apply Simpson's rule to correctly or not effectively, because you have to end with plus four to whatever the next to last one is, and then plus one for whatever your last one is, I'll call it C to the N. And in order for that to work, that last little number there needs to actually be an even number. This would have to be C14 or C12 in order for it to work. So you actually have to divide it in a fashion where you split it right up the middle. And you'll notice in this case, we didn't split it right up the middle. We actually hopped over the middle in there. So Simpson's rule, you have to be a little bit careful about that you need to have a correct number of divisions. In this case, we had an odd number of sections. You need to have an even number of sections to effectively use Simpson's rule. So a little more accurate, a little more work. But it's another pattern, one, four, two, four, two, four, two, four. And you'll notice because of the fours and the twos, they kind of average out to be three. And that's why there's this factor of three there as opposed to being a factor of two as in the trapezoidal rule. So hopefully you can now see how we've derived the trapezoidal rule and the Simpson's rule as means of calculating the area. We'll show a demonstration of this in a later video.