 So an important concept in higher mathematics is the idea of a region, and this is actually a fairly straightforward idea, and leads to the second problem of calculus, what's the area of a region? And one of the things about mathematics is that we can't ask a question without phrasing the question precisely. So we need to think about what we actually mean by a region, and so area only makes sense if we're looking at what are called bounded regions. So we need a region where there are boundaries, and conventionally we can view those boundaries as being a top part of the region, a bottom of the region, a left end point, and a right end point. So here's a region we have a top, bottom, left side, and right side. And it is only when we have those bounds that we can even ask the question, what's the area of the region? So we have to have some boundaries for the region. But once we have those boundaries, it makes sense to talk about the area of the region. So let's talk about some regions. So how about the area under the curve y equals 3x plus 2? Well, if I stop there, I don't have a region, because I have a top. I have an under this line, but I don't have a bottom. Under includes everything that's below, and that region has no bottom. I can't talk about an area. So I do need a bottom, so here maybe I'll use the x-axis. So now I'm under this curve, above the axis, and now I have this region here. But it doesn't have a left and it doesn't have a right. So it's not a bounded region. So it's not something that I can talk about the area of. So I have to supply a left and a right. So maybe I'll go from x equals 2 to x equals 7. And note that here x equals 2 forms the left side, x equals 7 forms the right side. And the grammar is it's over the interval from left to right. And that gives me a region whose area I can talk about. And just as a note, there's some other ways I can express this. So I might talk about this as the region bounded by y equals 3x plus 2 in the x-axis. And that statement bounded by says I'm not really going to commit myself to which of these two is the top and which of these two is the bottom. And that's perfectly reasonable because once we graph 3x plus 2, once we graph the x-axis, it's pretty clear that one of them has to be the top of the region, one of them has to be the bottom, and it's pretty clear which one is going to be which. And again, this interval from x equals 2 to x equals 7, I might just describe that as the interval 2 less than or equal to x, less than or equal to 7. Now this is the best possible way that you can have a region described to you because you have given explicitly what the top is, what the bottom is, what the left is, what the right is. But you might not always have that information. So I might talk about the area between the curves y equals 8 minus x squared and the curve y equals 2x. And if I graph those two curves, there is a region that is bounded by the two curves. There's this curve, there's this curve, and there is a region between those two curves, which I've shaded in red, and so it makes sense to talk about the area of that region. Now while the region makes sense as a description between these two curves, it's going to be a lot easier if we think about what our left and right boundaries are going to be. And where are those left and right boundaries? Well if I go all the way to the left, I see that one boundary is going to be where the two curves intersect, and if I go all the way to the right, I see the other boundary is also where the two curves intersect. So I need to find those intersection points, and I find that those intersection points are going to be located at negative 4, negative 8, and 2, 4. So there's my left boundary, there's my right boundary, there's my top, there's my bottom. And so I can say that the region goes from x equals negative 4 up to x equals 2. I'll have another example, how about the area between the curve y equals 16 minus 3x minus x squared, looks something like that, and the x-axis, I know where that's located. So again I have a curve, I have the x-axis, and I have a bounded region, which is going to be in between the two, right there, and as before, it's helpful to find where the right and left boundaries are, and again, way over on the left I have this point of intersection, way over on the right I have this point of intersection, and I can find those two points of intersection, and they occur at approximately x equals negative 5.77 and x equals 2.77. So I can make those two our left and right boundaries.