 Hello and welcome to a screencast about finding the mass of an object. Okay, today we are going to work on finding the total mass of a 2 meter rod that's lying on the x-axis with its left end at the origin. Okay, let me go ahead and draw that out. And then this rod has a density that follows the function 15x squared kilograms per meter. Okay, so my rod's going to look something like this. And we'll pretend like that's two meters long. Okay, so here's our rod. It's a constant with a constant area, constant whatever you want to talk about. But the density of this rod is actually changing. Okay, so let's go ahead and draw in our function so we can imagine what that density must be looking like. So here is our density function of 15x squared. Okay, so if we were to take a chunk of this rod, it would have a density, you know, kind of in the middle here. If we were to take something over to the left side, it would obviously have a smaller mass because it's got a smaller density than it would over here at the right side. Okay. All right, so then now we want to split up our rod. So as we've done here, and let's just take a look at one chunk. So what's the mass of this one chunk going to look like? Or one piece, I guess is probably a little bit nicer to say. Okay, so that's going to be approximated by our density function, which is rho of x times delta x. And the reason why that is is because of how the formula works out for mass is density times volume. Okay, so in this case we don't really have a volume here, but our delta x is going to give us the width of each of those chunks. Okay, so our density function then is 15x squared. So then our mass of one piece is going to be approximated then by 15x squared times delta x. Okay, so that's the mass of any random chunk along this particular rod that follows this particular density function. But I want to know what the total mass is going to be. So I'm going to need to be adding up then all of these different masses in order to get my total mass. Well, that idea gives us an integral, okay? Just like we talked about many times before in this chapter, you know, taking all these little pieces and adding them up is basically finding the area under that curve, which is the integral. Now how do I know what the end points on my integral are going to be? Well, I know that I'm going to be going with my delta x here, so I'm going to be going along my x-axis. So my x-axis goes from zero to two, so that's going to be my end points. And then I'm just going to integrate then this mass function. So that's going to be the 15x squared, and then my delta x then becomes dx when I go to do my integral. So this is a fairly nice function to integrate that ends up giving me an anti-derivative of 5x cubed. And I'm going to go ahead and apply the fundamental theorem of calculus, so throw it in your end points, throw in a two, throw in a zero, and you're going to end up with a grand total then a 40 for our mass. Now what are the units on this? Okay, so let's go back and look at the mass of one piece because if we can figure out what the units are of the mass of that one piece, then when we add up all the other pieces, they're going to have the same units. Okay, so our row function then has a unit of, this is going to be a big mass, let me just stick it right here. So we have kilometers per meter, and then our delta x, well that's measuring the little chunks that we're taking here of our rod, so that's obviously going to be measured then in meters. And this is kilograms, I think I said kilometers, sorry, I have meters on the brain. So this is kilograms per meter times meters. So those meters are going to cancel, so that's going to end up then giving us a unit of kilograms. So the total mass of this rod then is 40 kilograms. Alright, thank you for watching.