Alright, so I'd like to take a look at this guy here real quick. I'd like to find the moment of inertia of the rectangular sheet. Alright, and it's going to rotate around a line that has a maximal density, and that density is going to reduce. So I can't really draw that reduction in the density very well. I have ways of trying to do it, but they don't really look all that great. And shading and stuff takes a long time in doing this isn't the most amazing thing. So if this is, say, the wide direction, and this is the x direction, and so forth, and so on, rotating around that x direction, and right down here at zero, we have sort of the maximal density right there. Right? That's the maximum density. We'll call that maximum density sigma naught. Alright? And then when it goes all the way up to the maximum height here, H, the density is going to be zero. There's going to be, it's just going to taper off to zero. I don't know how that works. You could do this as an approximation for a very, for a thin wedge, for example, would work pretty well. Then I'm either side of the y axis, we go minus w over 2 and over here, w over 2 for the width. So we've got all the things that we really need, right in that description, or what we want to find, and we want to find its moment of inertia. So the first thing I always do is I just tally up what I know and what I want to find, right? So we want to, so if I've got a rectangular sheet. Right? That has a width of w, and a height of H, and has a variable density. I'm going to be, I should have said, with a variable density, but say, let me. And that variable density is going to be sigma of x, y equal to sigma naught. And we use sigma for the density if it's a surface density, if it's two dimensional density. You'll see a lot of that in the electrician magnetism class. So we'll do lambda for line density, sigma for surface density, and rho for volume density. Lambda and sigma make a lot of sex, right? Lambda is for line density is, or linear density, that's just the same, that lambda is L in Greek, right? And sigma is S in Greek. So those are making sense. So that row that doesn't make sense, but hey, not everything has to make sense. Right? Anyways, you already know about row, right? You've been using row for the density for a long time. Then you have this moment of inertia. Right? And that's I x. So it's rotating around x. We'll call it I x. And so that's going to be equal to in general, it's going to be equal to a double integral of r times sigma x, y, or just sigma, which is called sigma. So they're able sigma and then some distance, the distance from the axis to the point. So it's always going to be perpendicular to this rotational axis. And that's a square there. We square that distance for the second moment. Right? So it is a moment of inertia, but you use moments for other things too. And that's an integral over this entire area. So we have little area. That's a little area element dA right there. Okay. All right. So then we start filling these things in, right? Like the x stuff, that's minus w over 2 to w over 2, the y, that's 0 to h, right? 0 to h. And then sigma is sigma not 1 minus y over h. And this distance here is just the y coordinate, which is why I started at the zero in the first place is to make life easy for me. And then we have d y dx, right? So we're not getting anything really complicated. The x interval is comes out so we can pull out this differential element and just integrate minus w over 2 to w over 2 dx, which is just w, right? This minus that. We can pull out the sigma not because it's a constant. Then we have our integral 0 to h, y squared minus y cubed over h dy. All right. And these aren't, you know, sort of the world's most complicated integrals, which is, again, one of the reasons why I chose this. You know, it's not really the most realistic example. What it is is simple, right? So hopefully you can understand what I'm doing. So now I've got sigma not in w, I divide or integrate y squared. And that's one half or one cube, one third h cubed, right? And then we subtract zero from that. And so we don't have to worry about that. And then we subtract one quarter h to the fourth over h, which is still an h cubed. So, you know, nothing strange or mind blowing has happened. What else would we like to do? Well, now we can start pulling these things together, right? So this h cubed thing comes out. So we have sigma not w, h cubed. And then we have one third minus one quarter. And you'll remember that one third minus one quarter is something fairly simple called one 12. So we have sigma not w, h cubed over 12. And that is the moment of inertia for that thing. So that's reasonably simple. That doesn't make any, that doesn't make your life really, really hard. So I think that will, that will be enough for an example for these moment of inertia problems. I know in physics two, you didn't really do a lot of these integrals for moment of inertia things. And the ones you did were really quite simple. And so, you know, I just want to remind you about how this works because some of the moment of inertia things that they do in this book are a little bit complicated. So because it's a math class, basically. So they just want to give you most complicated things they can without breaking your brain. And you know, I think that's completely reasonable. I mean, that, they shouldn't want to break your brain. Right? So you should just be happy and thankful for that. Anyways, I will see you next time and have a great time. Bye.