 When welcome to this quick recap of section 6.3, density, mass, and center of mass. In this section, we'll be using some ideas from physics, including mass, which is the amount of matter contained in an object, and density, which is the distribution of matter throughout such an object. If we have a constant density, usually called by the Greek letter rho, then rho is equal to mass divided by volume, or m over v. Moving the v over, we get that mass is equal to density times volume, which is a key formula we're going to use. However, if an object has changing density, that is, it's not the same at all points throughout the object because there's more mass in some areas than the other, then the equation m equals rho times v doesn't directly apply. However, we can use one of our favorite techniques from this chapter, which is along an object such as this rod or piece of rebar or wire that we've drawn, we can imagine cutting it into many thin pieces. And each of those pieces can be approximately a constant density. So as long as we imagine that each of those pieces is close to having a constant density, we can use the formula m equals rho times v on each of those pieces. If you'd like, you can imagine the density as if it were a graph above this rod or piece of wire. The bigger the green box, the larger the density, and therefore, the larger the mass in that area. So to calculate the total mass of this rod or piece of wire, we can imagine taking the mass of just one piece and adding all of it up. So let's give some terminology here. We'll call the width of each piece delta x and the height of each box, rho of xi, that represents how dense the object is at that point. Rho is now a function, and it gives us different density values depending on which x value we put in. We want to add up all these pieces from the left end, which we'll call x equals a, to the right end, which we call x equals b. So to find the mass of one piece, we simply multiply the density, that's rho of xi, times the volume, which we're going to approximate by delta x in this case. To figure out the total mass, we will add up the density times the volume for all of these pieces. But whenever we have a sum of this sort, we should now say, let's think about taking a smaller and smaller delta x, meaning we get a better and better approximation. This is the exact idea we used when figuring out the area under a curve, and that led to a definite integral. And in this case, we get the exact same thing. Instead of multiplying density times delta x, we integrate rho of x, the density function, from our left endpoint to our right endpoint. This leads to the idea of center of mass. If you have several objects with masses that we'll call m1, m2, and so on, and they're distributed along a one dimensional object, an axis, such as a rod or piece of wire, at some positions that we'll call x1, x2, and so on, then the center of mass, or the balance point of this rod, is located at a place that has this formula. Effectively, we're multiplying the mass of each object by how far along the rod it is and adding all those values up. That's the numerator. And then we're dividing by the total mass, the sum of all of the masses. Again, this only works if we have constant masses at individual locations. If we were to imagine a thin rod or a piece of wire whose density varies from place to place, we can still imagine slicing it into thin pieces and pretending that each of those has a nice constant density so that we could calculate its mass. Again, we can imagine a mass represented by these green boxes sitting at each of those places, and we could use the formula that we just had to tell us where the center of mass is located. The formula turns out to look like this, and although it's an intimidating looking formula, each part is something we understand. This x1 represents the location of this first slice. The row of x1 represents how dense that particular part of the object is, so the value of the green box is height, and delta x represents the volume. So we're really multiplying density times volume, and that gives us mass, and then we're multiplying it by its location, the x1. So this formula on top is adding up masses times locations, just like our previous formula did. The denominator has density times delta x again, and so we're really adding up the masses of each individual piece. Take a moment and go back to compare this to the formula for the center of mass on the previous screen. Again, if we let delta x go to zero so that we get a better and better approximation for the center of mass, this turns into a definite integral. And in this case, the definite integral is for x times the density, because we're multiplying location by density, and the dx represents the volume units here. We divide by the integral of row of x, which is exactly the formula for the total mass that we had before. So here we're dividing the weighted location of the masses by the total mass. Now that we've seen these formulas, let's take a look at a few of them in action.