 Because the definite integral is the limit of the Riemann sum, then anything that we can express as a sum can be evaluated as an integral. And remember that the notation for the definite integral is actually an 18th century script S. And this leads to some important ideas. To begin with, here's an important concept. A variable quantity is nearly constant over a very short interval. Now this profound bit of wisdom suggests the following strategy. If we want to find a quantity, let's find a representative value over a short interval, and then sum the representative values. For example, here's how I got through physics. Let's say the density of a wire at a distance x from the end of the wire varies according to some formula. And let's see if we can find the mass of a 10 meter length of wire. So our units are helpful for figuring out what we need to find. The units of mass are going to be kilograms, and we note that our density is measured in kilograms per meter. So if we multiply this density by something measured in meters, we get an amount in kilograms. Now because our density is variable, we need to take a short length of wire and we can find its mass as density times length. So we'll multiply our density, given by our function, by our length, which, since x is our distance, our length is going to be a small bit of distance which we'll represent as dx. Well that's the mass of a short length. We want to sum up those masses from where we start to where we end. And since we're looking for the mass of a 10 meter length of wire, that means the distance from the end of the wire is going to vary from 0 to 10. And we can evaluate this definite integral.