 So the last of the three common types of integrals that you can see in fluid mechanics is that of a volume integral. We've looked thus far at line integrals, surface integrals. Now we're going to take a look at volume integrals. And these are quite common in control volume analysis. Surface integrals and volume integrals are common in control volume analysis. But let's assume that we have some volume, volume V. And what I'm going to do, I'm going to draw a little differential element in volume E of V. And we'll give it capital DS. And let's assume that there's a unit vector that points outwards and perpendicular to that volume V, or to that area, differential element. And this is a closed surface. So that means that it would be around the volume. It could be a cube, it could be a sphere, any kind of arbitrary shape. But when we're dealing with volume integrals, really, there's just two different things. One is a volume integral with a scalar. And the other one is a volume integral with a vector. And so we'll begin with the scalar. And for volume integrals, we show three integral signs to denote the three dimensions for volume. And given that we're closing a volume, I'm going to put the little circle through there. You don't always have to do that. But you can do it to make it nice and neat. The V denotes the fact that we're integrating over the volume. And let's say one that we see in the continuity equation and control volume analysis quite often is rho dV, which that would then give us the mass of a particular element. But what this is, so that's if you have some arbitrary scalar that you're integrating over for your volume. And the result is going to be a scalar. And then we can also have a volume integral where what we're integrating is some vector. So let's put an arbitrary vector here. So what we see is that when we do this, it's going to result in a vector. And those are the two different types of volume integrals we can encounter in fluid mechanics. So that covers most of the vector operations. The last thing we're going to do, we're going to look and see, there are certain relationships that link a line integral to a surface integral to a volume integral. And sometimes depending upon the nature of the data that you have, either from an experiment from numerical modeling or from analysis, you can go between lines and surfaces and volumes. And so that'll be the last topic that we look at as we review vector operators in fluid mechanics.