 So calculus is based on three important problems. We've seen two of them. Finding the slope of a tangent line and finding the maximum or minimum value of a function. Now let's take a look at the third of the three important problems of calculus. And that's this problem. Find the area of a bounded region. Now this term might be new, so for now let's define a bounded region as any portion of the plane that has an identifiable top, bottom, right, and left. For example, the region bounded by the curve y equals x squared, the x-axis, and the lines x equals 3 and x equals 5. Note that in this case, our top, bottom, right, and left are clearly identifiable curves. Yes, mathematicians use the term curve to apply two straight lines. However, it's not necessary that these all be curves. For example, we might have a region bounded by three curves, y equals x, 8 minus x, and 2x plus 5. Or we could have a region bounded by two curves, y equals x cubed minus x, and y equals x. Or even a single curve can bound an entire region, x squared plus 4y squared equals 16. So if I want to find the area of a bounded region, one thing we can do is to find bounds for that area. So I'll introduce the following terms. u is an upper bound for the area if the area of the region is guaranteed less than or equal to u. Likewise, l is a lower bound for the area if the area of the region is guaranteed greater than or equal to l. The closer u and l are to each other, the more precisely we know the area of the region. For example, let's take that region bounded by x squared plus 4y squared equals 16. And let's see if we can find an upper bound for that region. Now, since area is a geometric concept, the hardest way possible to answer this question is to not draw a picture. And indeed, answering a question like this without drawing a picture is a good way to gain street cred among bands of rogue mathematicians. But if you're not looking to impress bands of rogue mathematicians, we might want to answer this question using the easier way, graph the region. Once we have the region, it's a lot easier to figure out an upper bound for the area. Remember, the upper bound is going to be an area that the region is guaranteed to be smaller than. So let's throw down a larger figure. And if we can calculate the area of this figure, then we know an upper bound for the area of the region. And that's the problem. We have to know the area of the enclosing figure. So instead of using this figure, whose area we don't really know, we might use a different figure whose area we do know. So let's use a rectangle. We'll throw a rectangle down, and the area of this region is certainly less than the area of this rectangle. But maybe we can do a little bit better. And this rectangle seems to be a good fit. The entire region fits inside a rectangle that's 8 units wide and 4 units high. So 8 times 4, or 32, is an upper bound for the area of the region. How about a figure bounded by two graphs, y equals x squared and y equals 2x plus 3? Again, it'll probably help to draw a picture, so we'll sketch those two graphs. And we see that our line intersects the curve, and so we should find the intersection point. Since y equals x squared and y equals 2x plus 3, we can find the intersection point by setting x squared equal to 2x plus 3, and this gives us an equation we can solve. Getting all of our terms to the left-hand side, we see this is a quadratic equation, and so we can solve this using the quadratic formula. And we get x equals 3 and x equals negative 1. So we could try to find a box that the region fits into, and at this point, there's not really a whole lot of advantage in trying to get things too exact. So let's throw this into a fairly large box that has a width of 4 at a height of 9. And so this will give us an upper bound for our area of 36. Now, what can we do to get a better upper and lower bound? Well, one possibility is the following. Suppose I pick some points along the boundary of our region and then form a polygon by joining the points. The area of the polygon will approximate the area of the region. Actually, finding the area of a polygon is difficult, so in practice, we'll use rectangles because those are easier to work with. Now, before proceeding, we'll have to introduce a grammar lesson. Given a bounded region, we want to partition it into subregions. For convenience, for now, we'll identify partitioned points on the x-axis, and then use vertical lines through these points to partition our region into subregions. And for each of our subregions, the lower rectangle is the largest rectangle contained by the subregion. The upper rectangle is the smallest rectangle that contains the subregion. The left rectangle is the largest rectangle whose left side is in the subregion, and the right rectangle is the largest region whose, wait for it, right side is in the subregion. Now we have a whole bunch of rectangles whose areas we can determine exactly, and so we'll use our partition of the region to approximate its area. If we use the lower rectangles, we'll obtain the lower sum, which will be a lower bound for the area. On the other hand, if we use the upper rectangles, we'll obtain the upper sum, which will be an upper bound. We'll take another step that'll make our lives easier. Since using either the lower rectangles or the upper rectangles requires us to identify for each subregion, the lower or upper rectangle, it's easier, but somewhat less informative, to find the left sum found by using the left rectangles and the right sum found by using the right rectangles. We'll take a look at these next.