 So, in this video, we're going to take some time to calculate the area of a somewhat irregular shape. This irregular shape is the side of a Tian Deng flying lantern, and we're interested in finding the area of this shape because we're interested in figuring out how much weight there is per area of the paper casing that goes around the Tian Deng. However, this is not a standard shape. We can't use area equals length times width. We can't use area equals pi r squared or any of our standard formulas because we have a relatively irregular shape. So, we're going to use a method called numerical integration, particularly something called the trapezoidal rule, where we will approximate this shape by slicing it into each part and then do a pretty good approximate calculation for each of these slices. Now, you'll notice over here this sort of pedal shape that we had. This pedal shape, if we take it and we slice it, which is a measurement you've hopefully done earlier if you are doing this full analysis, if we slice it into slices here, each of these slices is spaced by roughly the same amount. In this case, these are all 20 units, and then this last increment is a little bit different. It's at ten and a half. But if we look at each of those pieces there and we kind of take a look at one of these, if I take that piece and I connect the points where those slices connect and I draw across the bottom here and I draw across the top and then even though these are curved, I'm going to draw a straight line from the two points on either side here and I'm going to notice that that shape is a trapezoid. Well, that trapezoidal shape isn't the perfect estimate of that shape there, but it's pretty good. It just misses a little bit of that curvature. If I wanted to get more accurate, I could divide that in half and then that little extra bump there would create slightly different trapezoids that would be better approximations. Then I could do that again, divide and divide, and I can make it as small as I wanted and each time I would get a little closer to the truth of that curve. But one of the tenets we use for numerical computation is more accuracy equals more work. Every time I divide those in half, that gives me a whole new set of data points and a whole new set of calculations that I'm doubling the number of calculations each time. So I have to decide some balance between the number of measurements and calculations I do and the level of accuracy I expect. I also have to realize that some of my measurements that I took here only have a certain level of precision, so getting too intense on how much calculation I do might just be wasted with the amount of precision that I'm taking here. In your examples, if you're following along with this set of lessons, you should probably have about twice as many here. I'm just keeping my numbers a little bit smaller for this demonstration. Also, we're going to do all of our calculations here by hand. However, doing those calculations by hand, we could actually probably simplify. We have columns and rows here. Generally when we see a process that's repeated over and over again and we are organizing our stuff in rows and columns, that would be a really good exercise for a spreadsheet. So let's talk about how we're going to go about doing this. Let's consider this, take this trapezoid and move it over here for a second. Here's a trapezoid and let's talk about what we know about finding the area of a trapezoid. Well, if you look at a trapezoid, any trapezoid like this, and if it's a regular trapezoid, which we're going to assume these are pretty close to being, you can usually take off this part, slice it over there and put it over to the other side, looking something like that. So we see that we have something that's somewhat like area equals base times height. Well, officially, the formula of area for a trapezoid, area for a trapezoid, is going to be equal to the height. We are going to have some perpendicular height here. There's a height there. And then we're going to have some combination of the two bases. Let's call this base one and call this base two. Well, in that particular case, if we take the average of the two bases, one half base one plus base two, notice I just put the one half underneath the height, there's our formula for the area of a trapezoid. We take these two and add them up, divide it by two, so we're averaging them. So it's basically the average width times the height, and that will give us the area of our trapezoid. Well, in order to do that with our data here, notice we're going to need a couple other columns that represent, well, first of all, this height from each of our trapezoids, which we said is pretty consistent, although it's a little different with the last one. So we're going to calculate it for each one. So we're going to need a value for our height, and we're going to call this the change in height because we've already called this h here, the height between each one. So our change in height between each of these is going to be important. And then we're also going to want something that we're going to call this one half base one plus base two, while our bases are just these widths we've measured. So I'm going to label that as w bar, meaning an average. It's a pretty standard notation for an average, so I'm going to go ahead and make a column for w bar, sort of this average width that's in between each of these. All right, so now that I'm going to do this, I'm going to need to have some extra columns to sort of put into place. So I'm going to go ahead and erase this picture here, and then we'll get back and put in the columns to calculate what do I need here? So we're looking for our delta h and then our w bar. Okay, so now I'm ready to put together my two columns here because I'll use those two things and multiply them together. Remember, my delta h times my w bar is what my area is going to be. So that's the final thing that I'm looking for is there's the one half base one plus base two and there's the height of each trapezoid. So I'm going to go ahead and calculate each of these. Now notice, and this is where it can become a little strange if you're using a spreadsheet, that the values we're calculating come from a connection between two subsequent rows. So as I write these columns, if I'm writing it on a piece of paper, I can kind of write it in between so that it represents a relationship between this piece and this piece or this piece and this piece. In a spreadsheet, you can't really do that. Everything stays lined up, so you have to kind of keep track of what you're doing. What I like to do when I'm doing that in a spreadsheet is even though it's on the same line, I will often use color in the column to indicate that this column is shifted a little bit, but here we can just write it up so it's shifted. So notice I go 20 minus zero. That's the change in height along this first stack. 20 minus zero, so that's 20.0. Similarly, 40 minus 20 is 20.0. And again, notice I'm writing it kind of in the middle of my rows here. Again, 20.0, 20.0, and then my last one is only 10.5. Now if we were doing a true trapezoidal rule in math, we would try to make sure that these were perfectly even, and then we could use sort of a longer formula that didn't take into account that one of these was different. But we're going to go ahead and just calculate each slice separately. All right, so now what I want to do is find the average between the two that are here. Each one of the ones that are here, so I'll have to add these two things together and then divide by two. So 70, that looks like 77.4, so that ends up being 37.5 plus two more, 37.7, I believe is the average of those two values there. And notice again, this is something that's calculated in between each of those parts. So I'm going to go ahead and sort of write it in a staggered row. So I'm going to go ahead and calculate the rest of these. This looks like somewhere between 51, this is 51.3, and this is going to be 56.3. This one's a little bit harder, 80, 90, 4, 95, I believe that's going to be 47.5, and last but not least, 19.3. Give me one second, I'm going to check and make sure my numbers are right. All right, and I was right, there is an error in here. Some of you following along may have found out that this is actually supposed to be 38.7, and there we go. So there's our value there, I did make a small error in that particular case. So now what do I do with each of these values? First of all, let's go ahead and recognize that these units are still in centimeters. We were adding and subtracting units in centimeters, so we get values in centimeters. And now I can simply take each of these values and multiply them. And what does each of those values represent? Well, it represents the area of one of my trapezoids. So in this case, let me do all the multiplications, and I'll go ahead and write them out here. Okay, so I've recorded all my area values here, 774, 1026, 1126, 950, 203. Notice I've also paid attention to how many significant digits I have. In each of these cases, I appear to have three significant digits. So as I move forward here, I keep three. Here it looks like I have four, but I like to follow a rule that if my first digits are one, if we just roll over and all my other multiplications that are similar are still in that, that I often treat that as a single digit. So we end up with a similar level of uncertain digit here. So there's three digits, three digits, three digits. And then this last one, because I was not multiplying by 20, but I was instead multiplying by 10.5, I got a value in my calculator of 202.65. Well, notice I could write that .65 down. Many high school students would, but then I would have this .65 added on when none of these actually have that level of precision. So I went ahead and used the appropriate number of significant figures here, which is still three. And we write round that up to 203. So now what do we have? We have one, two, three, four, five sections for which we've calculated the area of each. Notice this one is particularly small, not only because it goes to zero at that point, but it was only about half the height in the first place. Do you notice we do count that zero in there because that sort of accounts for a triangle. Remember the area of a triangle is one-half base times height. Well, remember that base, the second base is zero, so that's where that other base goes, and you still have the same formula for a triangle. It's just specialized to one of the sides being zero. Now, what do I do with these? Well, first of all, let's talk about the units. These are all in square centimeters because I did a centimeter times a centimeter. They're all units of area, which is what I would expect. And then finally, I total these to be 4,079 square centimeters. And that would be the area of one side of my lantern, noting that there are four of them. The surface area of the lantern, surface area of the entire lantern, would then be, let's see here, what we've got, 16,316 square centimeters for the surface area of the entire Tian Deng, using my numbers here. Notice then, with that surface area, if we were interested in sort of having a density value, not a true density because we're talking about density per unit area, not density per unit volume, but if we wanted a value like that, we could then take the mass of the paper and divide it by that surface area to get a density value for our paper that we could then use later on if we were building something to estimate the mass of the paper we would be using.