 Let's do a quick review of the main ideas of section 4.2 on Riemann sums. We saw in the last section that one way to estimate the distance traveled by a moving object, if we know its velocity function, is to construct a sum of rectangles by splitting up the time interval into a bunch of little pieces, making each rectangle go up to the velocity graph at the left endpoint, finding the areas which are actually little bits of distance traveled, and then add all of this up. In this section, we want to generalize this process so we can use it in a greater number of settings. For example, there's no particular reason to choose the left endpoint of the subintervals here instead of, say, the right endpoint, or the midpoint, or a randomly selected point within each interval. We could pick any point we wish in each interval, send the points up to the velocity graph, make the rectangles have those heights, then add up the areas. And we should still get a reasonable approximation to the distance traveled. In fact, the more rectangles we add, the better the estimates get, and the less it matters which point we started with in the subintervals. Here's a general construction that captures this process. We start with an interval on the x, or input axis. Let's say it goes from A to B. We pick a positive integer n and split the interval into that many subintervals. The width of each of those subintervals is the total length of the interval divided by n. We call that subinterval length delta x. On each subinterval, decide how you want to select the point. Usually we choose the left endpoint, or the right endpoint, or the midpoint. Let's say we pick the left endpoint. Label the endpoints of each of those intervals, x sub 0, x sub 1, and so on, up through x sub n on the far right. So the left endpoint of the first subinterval is x sub 0. The left endpoint of the second one is x sub 1. And in general, the left endpoint of the ith subinterval is x sub i minus 1. Send these points up to the curve and make rectangles on each base that are this high. The total distance traveled is estimated by the sum of the areas here. And each area is base times height. The base is always delta x, and the height is f of x sub 0, f of x sub 1, and so on, all the way up through f of x sub n minus 1, because we are taking these left endpoints and evaluating them into the function to send them up to the graph. Notice x sub n is not a left endpoint, and so it is left out. The resulting area estimates the distance traveled, and this sum is called the left Riemann sum, denoted l sub n because we used n subdivisions and chose the left endpoints. We can write this Riemann sum in sigma notation as follows. Sigma notation is a compact way to write sums, and there will be a screencast on reviewing how to work with sigma notation in general. We might also choose right endpoints instead of left, in which case we would have the right Riemann sum. Or if we use the midpoints of the intervals, this would be called the middle Riemann sum. In the section and in the screencasts, our goal is to set up and calculate Riemann sums and use them to estimate the distance traveled by a moving object given its velocity function. By the remaining screencast, we'll go into this in some depth now.