 So, today what we're going to do is we are going to apply our calculation known as numerical integration to determine the area of an irregular shape. And in this case, this irregular shape is a bird wing. So here we have a bird wing, the wing of, and I believe this is an Akkadian fly catcher if I remember correctly. So this is a wing that's been taken, and not only have we taken a picture of the wing, but we've taken a picture of the wing and we've interposed the picture of the wing on a grid to make it easier to do the math here. And I've already done the first step here and then I've decided that I'm going to go ahead and divide this wing into vertical divisions. I've taken each of the wing pieces and I've sliced it up into vertical slices along each part. You'll even notice that I've made some vertical slices over here on the right hand side, where I've sort of either sort of ended at a tip or sliced through the places on the wing and made a series of little segments over there. And I have to make an executive decision here about how I want to handle my slices at the end. And I'm going to decide that there is actually going to be a place over here where we are going to say, okay, wherever there is a spot there, we are going to assume that there is a place here out at my very end and that we're going to approximate the wing tips of the bird as extending out to that spot there, that my ending point is actually not going to be this line, this set of lines here, but is instead going to be a zero point or a set of zero points out here. And that's going to be part of my approximation. All right. So now that I've done that, I've also need to determine what my units are going to be. I've decided that my unit in this particular case is going to be one of these little grids between the red and the blue. There's one unit for my unit purposes. Now you will notice that I have a scale down below, a centimeter scale down below, and I could sort of start right now approximating with that centimeter scale. But I'm not going to do that at the moment. I'm going to go ahead and use my grid units for my calculations, and then I will do a conversion later to determine what the actual areas in terms of real units, in this case centimeters. All right. So to do this process, we're going to need a series of measurements. The first measurement that we're going to need is the measurement across each of these units, okay, along that span of the wing. So that little piece there, we're going to recognize that that is two units. And that, I think we called that a piece of our span, little delta s, sometimes used w as a width or whatever. So each of those pieces, we're going to say is two units. And then what we're going to need for each of those things is then we're going to need to go ahead and measure each of the lengths along these vertical divisions. So I'm going to count each vertical division here, and I'm going to look and say, okay, let's see here, one, two, three, and a little bit more. So I'm going to go ahead and make, for each of these vertical divisions, I'm going to go ahead and make some measurements. And right down, record, in this case, I see this as being 3.3 units as that very first one. And I'm going to go through each of these lines and record what I've already, I've already looked at these carefully and measured them. So I'm going to go ahead and record these now along the screen, and you can look at them carefully yourself. So let me go ahead and record the different unit measurements for each of these, 11.7. Now you'll notice for this last two, the 4.3 units is not the distance from the top to the bottom. It's actually sort of what I estimate is the sum of all these pieces, as we're going to consider this to be a series of little trapezoids that go between each of those wingtips. And then I've decided to use zero as an actual point as well, which means that what I'm choosing to do is estimate the tips of the wings as being little triangles that come down to a point there. And that's a choice that I have to make, whether to use zero there or not to use zero there to take into account, to approximate these wingtips that have been sliced off, whether we're including them as little triangles or not. I've decided to count that number zero as one of my lengths. So now that I've got all those measurements, I have to go and recall what the trapezoidal rule looks like. And if I remember, the trapezoidal rule looks something like this. A, my area, is equal to 1 half times that thickness, which we call delta S. So there's that delta S thickness plus 1 times my first chord. So that would be C naught plus 2 times my second chord plus 2 times my third chord plus 2, et cetera, et cetera, plus 2 times my next to last chord, which is how many? 1, 2, 3, 4, 5, sorry, 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11. So 2 times C 10 plus C 11. Now you'll notice that that last one, that 11th one, is actually zero. So did I even have to write that? No, I didn't. But the reason to include it is you'll notice the 2 that gets counted here. That's C 10 instead of being just counted once is being counted twice because we're counting the zero as one of our measurements. So now I notice there's a pattern there, which is 1, 2, 2, 2, 2, 2, 2, 2, 2. So I'm going to say, OK, let's go to each of these ones that's in the middle. And we're going to take each of those values and double it. All of these middle values need to be multiplied by 2, including that 4.3 that's there at the end. We're going to multiply all those middle values and record them. So I'm going to go ahead and do that now. And then the units on the end, I'm going to keep the same. So now that I've done that, I'm following my pattern. I can put all those things in together. And I'm going to take them all and add them all up. So my area is going to be 1 half times my delta S of 2 units, 2 units, which very handily is going to just cancel out the 2s in that case, times the sum of all those multiplied or what the term is weighted. They've all been weighted. They've even counted once or they've been counted twice. And that term is called weighted. Once I've weighted all of those, I'm going to add them all up. I believe, if my math is right here, that they add up to 188.4 linear units. And once I've actually done that, my 188.4 units, we'll notice this is pretty straightforward. The 2s cancel out and I get an overall area of 188.4 square units. Hopefully you can see the math there. Not too difficult. So I'm not quite done yet, though. One thing I have to recognize here that these are grid boxes. And those grids were kind of arbitrarily placed, depending on the graph paper that I happened to have around and how I laid that wing picture onto it. So now I need to go ahead and figure out what this is in real units based on the scale that was provided by my wing. So there I'm going to look carefully and I'm going to recognize and I'm going to look at this whole scale and I realize that this piece here is 1, 2, 3, 4, 5 centimeters. And that five linear centimeters are the equivalent of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are the equivalent of 10 of my units. OK, well, now I can make a conversion. Let me go ahead and rewrite that as 1 centimeter is equal to 2 of my units. Well, my area has been calculated in units. So now I'm going to convert that area from units into centimeters. So I'll draw a little line here and use a ratio. I recognize that two units corresponds to 1 centimeter. But here's where many students will make a common mistake. They will divide by 2 and then they will say they're done. However, this measurement over here is a linear measurement. It's a conversion to convert lines to lines. Notice I'm a two-dimensional area measurement. If I take this same centimeters and I make it a 5 by 5 square, I can't quite fit the 5 by 5 square. I guess I can fit the 5 by 5 square in there. If I took that same measurement, the relationship between linear and squared is not the same. Because we have two dimensions, we have to do this conversion a second time. In other words, well, basically, we're doing the conversion a second time. So we're going to end up with units we'll cancel out. We'll end up with an answer in centimeters squared. But now it's 188.4 divided by 4. And that 188.4 divided by 4. Let's see here, 188.4 divided by 4 equals 47.1 square centimeters. And that's the area of our Arcadian fly catcher, Erdwing.