 Let's do a quick review of the main ideas of section 4.3 on the definite integral. So the last couple of sections we have been considering the question of how to find the distance traveled by a moving object if we know its velocity function. One way to answer this is with a Riemann sum, which involves subdividing the time interval, sampling a point like the left or right endpoint from each subinterval, making a rectangle with each point, finding the area of each rectangle and then adding up all those areas. One of the things you probably noticed is that a Riemann sum with a single big rectangle is easy to set up, but it's a terrible estimate in all but the most trivial of cases. On the other hand, if we sample more frequently and use a Riemann sum with a large number of rectangles, then this is more computationally intensive. We have more work to do, but we can see that we're going to get a much better estimate. We get more rectangles by increasing the number of subdivisions of the time interval. We denoted that number by n. So as n increases without bound, the Riemann sum gets better and better as an estimate. Therefore, the limit of this Riemann sum as n goes to infinity would be a quote unquote perfect estimate. In other words, it would be the exact value of the area under the velocity curve, otherwise known as the exact value of distance traveled. This limit of a Riemann sum is what we call the definite integral of the function f on the interval from a to b. We denote it here with the symbol, which is supposed to be evocative of an elongated s for some. The a and the b here are called the limits of integration, and the f is the integrand. Finding out the value of a definite integral is called evaluating the integral. So how do we evaluate a definite integral? It looks as though evaluating a definite integral involves evaluating a limit. But in fact, in some cases, evaluating an integral is quite simple if the graph is highly geometric. For example, if the graph of f of x is composed of line segments and circles, then the definite integral can be evaluated exactly using geometry formulas, because the value of the integral is the exact value of the area between the graph of f and the x-axis. Elsewhere in this section, we discuss some basic integral properties, such as the definite integral from a to itself is zero. A definite integral can be split up along the interval we are integrating over. And we have a constant multiple rule for definite integrals and a sum rule for integrals. Finally, we introduce the notion of the average value of a function on the interval from a to b, which is defined to be the definite integral on a b divided by the length of the interval. This is not the same as a statistical average because f is not a discrete set of disconnected data points, but rather a continuous function that is in constant motion over the interval from a to b. So basic stats won't help us find an average, but the integral can do it.