 Hello, and welcome to this screencast on section 11.1, double Riemann sums over rectangles. In single variable calculus, recall that we approximated the area under the graph of a positive function F on an interval from A to B by adding the areas of rectangles whose heights are determined by the curve. The general process involved subdividing the interval A to B into smaller sub-intervals, subtracting rectangles on each of these smaller intervals, then summing the areas of these rectangles to approximate the area under the curve. In this screencast, we will extend this process to two variable functions over rectangular domains. Since we are dealing with functions whose graphs are in three dimensions, the area under a curve is no longer relevant here, instead we aim to approximate the volume underneath the surface generated by the graph. To do this, we will start with domains that are easy to handle, rectangles. Let F be a continuous function of two variables on a rectangular domain R. Note here we are only displaying the set of inputs to the function F. We see that the X values are between A and B, and the Y values are between C and D on this domain. Here's a picture of what R could look like in the XY plane. Next, we do a similar thing as what we did in single variable calculus. We're going to partition the intervals from A to B and from C to D into sub-intervals. And using those sub-intervals, we create a partition of the rectangle R into sub-rectangles, as shown here with the gray lines. These sub-rectangles have length denoted by delta X, width denoted by delta Y, and area equal to delta A, which is the product of delta X and delta Y. To keep track of all the different sub-rectangles, we typically give each sub-rectangle two numbers, one number for the X sub-interval and one number for the Y sub-interval. Here you see that we've labeled the X sub-intervals starting at 1, and then the next one is 2, 3, 4, and the Y sub-intervals are labeled similarly, the first one starting at 1 and then 2 and then 3. And then if we want to pick out a particular sub-rectangle within R, for instance this one, it's labeled first by the X sub-interval label, and then next by the Y sub-interval label. So next we're going to choose a point in each sub-rectangle. The point labeled here comes from the sub-rectangle labeled 3, 2, that we just looked at on the previous slide. So we label the subscript of the X and Y coordinates accordingly. Using this point, we plug this in to get the height of a rectangular box. This rectangular box has length delta X, width delta Y, and height given by the value of F at the point that we chose. We can write the volume of this box as a product of these three dimensions. And then recall that we can write this more succinctly using delta A since delta X times delta Y is equal to delta A. Right, so this gives us the volume of a rectangular box for this particular sub-rectangle. If we do this for all the different sub-rectangles that we see here to get rectangular boxes for each, we can use the volumes of all these rectangular boxes to approximate the volume between the surface and the XY plane. For instance, suppose we wanted to use this procedure to get the volume displayed here under this function F, so between F and the XY plane. Here's what our rectangular boxes could look like in blue. You see that the sum of the volumes of these boxes approximates the volume underneath F. Now that we've seen what these rectangular boxes look like, how do we add up the volumes to get our approximation? Well, we take the volume of the first rectangular box plus the volume of the second plus the volume of the third. And we keep summing until we get to the end of the first row of boxes. Then we add up all the volumes from the second row of boxes and the third row of boxes. And we keep going and we stop once we've gotten to the last row. There are a lot of terms in this sum. This large sum that we're looking at is called the double Riemann sum for the function F over the rectangular domain R and can be written more succinctly using two summations as we have here. Okay, so now using this double sum, as we increase the number of rectangular boxes without bound, our approximation gets closer and closer to the actual volume underneath the surface, as we see here. This is a limit of the double Riemann sum that we were just looking at. And the value of this limit is called the double integral of F over the rectangle R as we have defined here. Now, we've already seen one interpretation of the double integral. The double integral represents the volume between the function F and the XY plane. Note that this volume is signed, meaning that any volume below the XY plane is counted as a negative volume. You will explore these ideas and more in our work throughout section 11.1.