 So, I've gone to a great deal of trouble to develop enough calculus to allow us to evaluate definite integrals, but before proceeding, let's see if we can try to find some definite integrals without using calculus. For example, suppose I want to find the definite integral from 0 to 5 of 10 minus 2x. The starting point for this is identifying the geometric meaning of this definite integral. And so we see that this definite integral corresponds to the area that is under y equals 10 minus 2x above the x-axis from x equals 0 to x equals 5. Let's sketch this region. First, we want to be under the graph of y equals 10 minus 2x above the x-axis starting from x equals 0 and then they get x equals 5. And so this is the region that corresponds to the definite integral. So if we can find the area of this region, we can find the value of the definite integral. And that's easy enough to do. This region is a triangle with a base of 5 and a height of 10, so its area is going to be, which will also be the value of the definite integral. Well, how about a different one? So this integral corresponds to the area of the region, but this region consists only of the line segment, so the area is going to be 0.