 If we know the value of the definite integral and some of the properties of the function, we can find other information. These can be found from the properties of the definite integral as the limit of the Riemann sum. And in particular, if our function is positive over some interval, then the definite integral will be the area between the graph xf of x and the x-axis over the interval. For example, suppose our function is positive and increasing over the interval between 8 and 20, and for this suppose we know the value of the definite integral, let's find the greatest possible value of f of 8. So let's sketch a graph of y equals f of x, where f of x is positive and increasing over the interval. So the interval starts at x equals 8, runs to x equals 20, and f of x is positive and increasing, so maybe it looks like... And again, if it's not written down, it didn't happen, so let's go ahead and label. And the important idea here is that we're not going to read anything directly off the graph, but our graph will help us organize our reasoning. So let's think about that. We have the value of the definite integral, and since the value of the definite integral is equal to 100, then the area between the graph and the x-axis over the interval is 100. Well, we're interested in saying something about f of 8, so if we use n equals 1 left rectangles, then the left rectangle will have a height of f of 8 and a width of 20 minus 8 or 12. And since f of x is increasing, the area of this rectangle will be less than the actual area under the curve, and so we know that f of 8 times 12 is less than 100, and so we know that f of 8 is less than 112. Or I suppose f of x is positive in decreasing over our interval, and let's say we know the value of the integral from 0 to 10 and from 10 to 20. What can we say about the greatest and least possible values of f of 10 itself? So let's sketch a graph of y equals f of x, where f of x is positive and decreasing for all x. And again, if it's not written down, it didn't happen, so let's label. So this first integral from 0 to 10, well that's the area under the graph over the interval 0 to 10. We know that's 70. Let's go ahead and draw that area in and label. And since f of x is positive in decreasing, then the area of the region will be greater than the area of the right rectangle, and that right rectangle has height f of 10 and width 10 minus 0. And so this tells us something about f of 10, namely that it has to be less than 7. Similarly, the integral from 10 to 20, well that's the area under the curve, and we see that that area has to be less than the left rectangle with height f of 10 and width 20 minus 10. And that tells us something about the value of f of 10. And so what we know is that f of 10, whatever it is, has to be someplace between 3 and 7. Thank you.