Calculus: Integrating with Respect to y: Part I

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Applications of Integration
Area Between Two Curves Not in x
Finding Areas by Integrating with Respect to y: Part One Page [1 of 1]
Well, we’re having fun with looking at all of these areas bounded by exotic functions and whatnot and getting these exotic shapes. Now what I want to look at is what happens if the shapes are really exotic? So what do I mean by really exotic? Well, let me actually give you first of all a sketch – two sketches, in fact. Look at that! If you had those special 3-D glasses, you could see these things in stereo.
In this graph what I want to do is I just want to draw two functions. One looks like this, and one that looks like this. So these are two functions, and suppose that we’re given endpoints – I’ll do my endpoints in purple here – and we want to find the area of that region. Well, this is something that we talked about an awful lot recently. So we get a sense of these. We just take the top function and we subtract the bottom function, and that gives us the height of the generic rectangle. Then what we do is we sum those areas up basically from the first purple point all the way out to the second purple point. In practice, that requires us to take the anti-derivative. We evaluate the anti-derivative at the left, the right, and we subtract and we get the actual area – the area of the yellow.
Now let me just take a look at this, and I want to actually now show you those rectangles that we’re putting in. I could draw it, but I actually want to demonstrate it sort of live for you. In fact, here are the rectangles, and I hope you can see this. You see, here the rectangles are quite long, and then they move up a little bit like this. Notice that always they might get a little longer in the longer spots, and they get shorter in the shorter spots, but always you’ll notice that the rectangles have the same spirit. They start on the green and go until the orange. They are all like that, and that’s why we take the top minus the bottom.
Now what I’d like to do is…can you see that mark there? The prop person said that that would not come off of my finger, and I said it would, so for the records, if you take a rubber band and mark it with magic marker, it does come off on your finger. In fact, it comes off everywhere! So much for the prop department.
Now let’s take a look at the following curve. So here we see this scenario. Let me draw another picture for you. How about a picture like this? So there’s a curve. Let me draw another curve here. There’s another curve. Suppose you wanted to find the area bounded by these two curves. So I want to find just the area between them. So it would be all of this stuff right here – all of that yellow. How would we proceed? Well, we’d try to put those messy rectangles in, and we’d want to look for the shape – the generic shape of the rectangle – in particular, what are we subtracting from what? You can see that when I put the rectangle in right here, it’s definitely the case that I’m taking the green and I’m subtracting the orange. So you might be thinking green minus orange. But look what happens sort of when I get over to this area here. Well, this area actually requires me to take green – top green – minus bottom green. So somehow I’m subtracting the function from itself. Well, the problem is this is actually not a function. This fails the vertical line test. So this is not a function, this is not a function, so these rectangles are having a real hard time fitting into place because you see I’m going from the top of the graph to the bottom of the same graph. How do I subtract something from itself? Then I go out here, and then all of the sudden the orange kicks in on the bottom, and now on the top I have green, on the bottom I have orange, and then later in life I just have orange to orange. So, in fact, this is really hard because there are a lot of different shapes of this rectangle. There’s no generic rectangle. Sometimes the rectangles go from green to green, sometimes they go from green to orange, and sometimes they go from orange to orange. It’s sort of annoying because here always there was a beautiful simplicity here. I always went from green to orange, green to orange, green to orange. There was always a nice way of explaining and describing that. Here I see green green, green green; green orange, green orange; orange orange, orange orange. It’s very complicated.
However, notice that if you were to tilt your head 90 degrees like that, then I would be looking normal right now if you were really doing this. And if you now look down, what would you see if you put the rectangles in like that? Well, I would always have orange here, I would always have green here, and notice that is always the case. So this actually has a nice simplicity once you turn your head 90 degrees to the side. Do you see that? See how nice and simple that is?
So this is a problem that I should be integrating and summing, not like this – sum, sum, sum, sum of the rectangles; sum, sum, sum – but I should be summing up and down instead of right to left. So let’s think about that idea. With that idea classically, and in these examples that we’ve looked at, when I put in that generic rectangle, the rectangle we take top minus bottom, and it’s a small change in x. When I can stack the rectangles like this – like little soldiers, one next to the other and I fill up sort of space in the x-direction – I refer to this as an x-easy picture. (By the way, this is just my special notation.) But the idea is that I’m integrating and summing with respect to x; there’s a small change in x. Those are all of the examples we’ve been doing. But here I see that this is actually not x-easy – it’s not easy to put in those rectangles. But it is easy this way, so I should draw the generic rectangles that look like this because then all of the rectangles will have that same basic shape – either longer or shorter, but they always go from this green out to this orange. And if I stack this way, I see a small change in y, and the rectangles instead of being stacked like this, they’re stacked on top of each other and they’re stacked like dishes. I refer to this kind of stacking as y-easy. In this case, my rectangles would have area is base times height. Now the base is this long thing, and the height is a very tiny change in which direction? It’s a tiny change not in x, now it’s a tiny change in y. You see, there’s a little teeny change in the y-direction. Therefore, my integral is going to be with respect to y. So here integrals are going to be with respect to the y’s, and here integrals are with respect to the x. So the idea here is that if you can put in the rectangles in a nice uniform manner and you can put them in vertically, then that’s x-easy, which means you should integrate with respect to x. If you can’t stack them that way but they stack really nicely on top of each other horizontally – lying horizontally – then the small change is y, and you should be integrating with respect to y, and this is called y-easy.
So this is the basic idea of actually finding areas of things that aren’t functions but merely relations. Up next, I’ll actually work through a couple of very explicit examples so you can see this in action. I’ll meet you in the example section coming up next. I’ll see you there. Bye.