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This is a video on the area between functions.
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And when you're done watching this, you should have a good idea about how integration can be used to help you find the area between two functions and how to interpret that area.
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So let's start off with the interpretation.
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Here's a graph. The lower graph is the imports annual imports in Canada for the years 1997 to 2001.
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And the upper graph is Canada's exports over that same period of time, annual exports per year.
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So we know from previous videos that if we want to get the total value of all their imports from 1997 to 2001, we would simply take the integral here, the integral of the import function.
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We remember that's going to actually be per year annual ones times what? Times that's change in time, dt from what? 7, which is what I'm calling 1997 to 11, which I'm calling 2001.
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So this is the area underneath the eye curve, if you will, this obviously then is the area underneath the export curve.
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And if I subtract the imports, the total imports for those years from the total exports from those years, I'm going to end up with the trade balance, the total trade balance from 1997 to 2001.
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And graphically, when I subtract these two areas, what I end up with is the area in between.
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So you can see if I take the red area, the red shaded area, and I subtract off the blue shaded area, I end up with the green shaded area, which happens to be the area between the two functions or between the two curves.
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Turns out that I can actually subtract the two functions first and then take their integral.
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So I can think of this one of two ways when I'm actually going to compute it. I can compute each area separately and subtract them, or I can create the difference function, the difference function of the two functions, and integrate that over the same limit 7 to 11.
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And you're going to discover that this here, the green way, if you will, is going to be the fastest way to do this in your calculator if somebody gave you these two functions, E and I.
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Of course, trade balance can be referred to as trade deficit, or trade surplus, depending upon what, depending upon how if imports are bigger than exports.
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In this case, imports were always smaller than exports, so the total trade balance was always positive and therefore always a surplus.
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But let's take a look at another graph to see where that might not be so.
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Okay, so here's another country, but unfortunately what, the imports are above the exports for some of the time, and then thankfully at some point the exports rise up.
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So if I was starting to think of that total trade balance again, I would start integrating and therefore integrating the exports minus the imports.
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So I'm going to end up adding, computing the area of this, but something's happening here that wasn't happening in the last one.
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And that is that the two functions are changing roles here, here at first imports is bigger, but then later imports is lower.
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So it turns out if I'm going to subtract exports, if I'm going to take exports minus imports, if I'm going to take exports minus imports, and take the integral of that, then over in this region that function is going to be negative, exports minus imports because why?
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Because exports are smaller than imports, and if I'm subtracting it, I'm going to end up with a negative. And so it turns out that this area right here is actually a negative area, so I'm going to indicate that with red, whereas I'm going to leave this area over here as green, why?
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Because it's area is positive. So if I'm going to integrate, for example, from 1997 to where to 1999, I'm going to discover that the trade balance is negative. In other words, the total trade balance is negative. This area is negative.
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Therefore, it's a trade deficit. Okay, so what happens right here? Are they in balance right at this point? And whatever that is, mid 1999 is the trade deficit, all of a sudden become a trade surplus as I move beyond here?
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The answer is no, not the total one, because before I'm going to end up with a trade surplus, this area over here has to be at least as big as this area.
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So if I integrate from here, all the way over to here, all the way over to 2001, what I'm going to discover is that the trade balance is still negative. It's better, it gets better, beyond mid 1999, the trade balance, the deficit gets smaller and smaller and smaller.
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But when I get to 2001, it's still negative. Why? Because the positive area over here is still smaller than the negative area that we originally calculated up to mid 1999.
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Okay, so there's another conceptual idea about what's going on. Let's try another one. Here we have cost and revenue. Okay? Cost and revenue.
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So first of all, the cost and the revenue are what? They're daily, aren't they? They're daily cost and also daily revenue. And when we multiply by time, we're going to get the total revenue and the total cost.
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So the area under this curve, the revenue curve would be the total revenue and the area under this curve would be the total cost. And if I subtract those areas at any day, any day, then what I'm trying to do is find the total profit, the total profit.
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So what's been happening right here? The cost, the cost right here, the revenue, I guess, is bigger than the cost, right? The revenue is bigger than the cost.
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So should I mark everything that's a positive area in green? Sure. So the revenue is bigger than the cost. So when I go to take what? When I go to take the integral of what?
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The revenue minus the cost, then in this area, that revenue is going to be, or that profit rather, the total profit here. Let's call this piece of T here.
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These are daily revenue and cost, but this is a total profit here. When I get to day one right here, from zero to one, that's going to be a positive because why? Because R, the daily revenue, was bigger than the daily cost in that whole region.
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But then what? Then my total profit starts to go down as I add this up. And when I finally get in, no, I don't know, right about here somewhere.
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It looks like this area is now the same, just about the same as that area, maybe even close to day two. These areas are about the same. So I could probably say with good confidence that the profit, the total profit is pretty close to zero at day two.
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Because this area, the red area, is now canceling out the green area. And as I proceed forward, when I proceed forward, now my what? My total profit is definitely negative. I'm actually losing money. I'm continuing to lose money.
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Until I get to where, until I get to this point, day four, where now, what? Where all of a sudden, I actually start making money at each day, but I'm still in the hole. Why? Because this little area, plus this little area, is not as big as this area right here.
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So I still got to let more time go by before I'm going to end up back at zero profit. How far? I don't know. I guess I have to eyeball it as best I can do. So maybe like that. Does this area, and this area now about the same as that?
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If it is, then somewhere between day five and day six is when the total profit is back to zero. And then what? As I proceed forward, now my total profit is in the positive arena.
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So don't get confused. There's like daily profit, which definitely is changing at all these places. So here the daily profit is bigger, is positive. Right? The daily profit is positive here, for example, right here. The daily profit is positive, but not at all is the total profit from zero to four in a quarter.
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I say the total profit is still negative, even though the daily profit at day four in a quarter is positive. Okay, what else? Anything else here? I got another one orders in inventory. What does this kind of mean? Let's see.
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If inventory is bigger than orders, then what is that? Is that good? Is that good? I guess we'll call that good. Okay? We'll call it good. So we'll paint it green.
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Okay? And if what? And if orders are greater than inventory, then we're out of stock, right? We're out of stock. Okay? And then when inventory tops orders again, we're looking good. Okay?
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So let's see, which is the greatest from zero to seven, the whole thing? Orders minus inventory or inventory minus orders? Or the area between the orders and inventory curves? Okay? Well according to this, what? If I take orders, orders, which is right here, orders minus inventory. Let's see, as I did the green ones, what did I do?
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I did inventory minus orders to get the green ones and the red ones as I mark them. So like this one is going to be what? It's definitely going to be what? Bigger than zero? But this one is going to be what? Well that would switch them around the other way. So this would be my green area. And these would be to my two red ones.
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So this one's definitely going to be less than zero. So according to the sign of things, not the size, these two, this one is going to be bigger. But what about the area between the orders and inventory curves? Well that just means add up all the areas.
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So because I'm going to end up subtracting off this area from all the area, this number right here, B is going to end up being smaller than the total area. So technically the choice here is this one. Although this is not all that interesting to us, the area between the curves, what's really interesting to us is probably this one, the inventory minus the orders.
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Because we would like that to be what? On the whole, we would like it to be bigger than zero, the total inventory minus orders. That's telling us that on the whole, we've always had more in stock than what people were ordering.
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And we could take a look at this other question here. How about this one? Get this guy out of the way. Put this one in here.
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It says multiple choice. The answer to part A measures what? The answer to part A measures the accumulated gap between orders and inventory.
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The accumulated surplus through day three minus the accumulated shortage for days three to five. Blah blah blah blah blah. And let's see day three. Where's day three? That seems crazy. The accumulated surplus through day three minus the accumulated shortage for days three to five plus the accumulated surplus for days six to seven.
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I don't think so. I think two and five are what we want. The total net surplus. I think that's what we were trying to talk about right here or the total net loss. I think we're going to choose C here because the total net surplus means having more inventory than orders on the total.
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Let's go with C. If we want to refer to C would refer to B rather and probably the accumulated gap between orders and inventory. That would be just this total area which I said isn't all that interesting. What's interesting is B there.
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There's a bunch of different examples to get you on the ground. How about actually doing a problem now. Let's do this actual integration. We have revenue and cost. What do we want to do? We want to integrate revenue minus cost.
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We want to integrate revenue minus cost. We want to integrate from zero to five of daily revenue minus daily revenue minus daily cost.
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We're going to end up with what? We're going to end up with the total profit from day zero to five. That's why putting a sub T on it. These ones are daily and if it has a sub T I'm saying it's the total profit.
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That would be the area between the graphs RRT and C of T. Now we've got to be a little careful here because they said area. Do they mean that they want to find the net total profit because that's what this is.
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Hopefully that's what they're after. Let's take the calculator up and see what we can do. I'm missing my calculator RATS.
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Here comes the calculator.
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If we take a look at our problem we have 100T and then minus 90 plus 5T. In here I'm going to clear out whatever I got here. I'm going to do 100 plus.
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What was that again? 10T, 10X, minus and here's the tricky part. I've got to put a parenthesis here or else I'm not going to negate the whole thing. 90 minus 5T.
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90 was at minus or plus. 90 plus 5T. 5T which will be 5X here. I could separate those two. Remember I could do the integrals separately but if I subtract them then I'll be able to take the integral separately.
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I might want to look at this thing between 0 and 5 for example. Just to see what's going to happen. I don't know what my upper window is going to be. Let's see. It starts at 100.
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I'm subtracting off 90. Let's try minus 10 to 10 and see if that's big enough. Let's see what we get.
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We get a pretty boring line there. Hold on a second. I got this numerical derivative on here. I'm going to turn that off. Let's see.
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I'm going to start at 100. Let me take a look here. I'm going to subtract 90 so that's 10 and I'm going to have a slope of 5.
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I think my window has to go up a little bit. Let's make this 30. Let's try it. There. Now I'm seeing it.
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It doesn't go negative so I don't have to worry about this issue about whether they intended to be the total area or the net total profit since it looks like the difference of the two functions is always positive.
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At this point I can run the integral on it if I want to. Let's see. What are the ways? In the other video I showed you all the different ways you can do it. Here's the way you can do it if you're looking at the graph. Let's take advantage of this way.
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There's a solver method that you can use. If you want to from 0 to 5.
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Let's add it up and then tell me the area. The total profit is 112.
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There you go. So an introduction to the area between curves, what it means and how to compute it using your calculator.