 So let's look at a couple of examples of the area moment of inertia applied. Or actually, we're not quite going to apply it at this point, but we're going to calculate some of these values for the area moment of inertia. And both of our examples are going to consider the idea of a cross-section, where we have some sort of long beam or rod, and that beam has a cross-sectional area that is consistent, and that we're considering cutting across that beam to calculate the cross-sectional area moment of inertia. And there's a number of possible shapes. On the left here, I have an example of a box beam, which might have certain dimensions, B for sort of a breadth and D for a depth and a thickness T, or we might have a hollow circular rod with an inner and outer diameter and a particular thickness. We're actually going to look at examples of each of these, how you would calculate the moment of inertia for each of these using standard formulas. There are also calculus methods that you can use to apply to any shape or form, but that's beyond the scope of this discussion. So let's consider first a box beam, and this box beam we're going to give certain dimensions. We'll consider it to be six inches by four inches, and with a thickness of one-half inch, and the six inches will be the vertical dimension, and then you can see that we have a half-inch thickness all the way around. Now, if we were interested in the axial stress on a beam with this cross-section, we might care about the axial stress, which is equal to the force divided by the area. We could calculate the area using the geometry we have. But in this case, we might be interested in the bending, and if you remember, the stress due to bending, sigma equals MY over I, and that I in this case is going to be the second moment of area, or the area moment of inertia around the middle of the beam. So we're going to want to calculate this value Ixx to determine how well this box beam will resist a bending moment. So there are a number of resources we can use, standard, statics, and strength of materials textbooks have charts like this. There are plenty of resources on the Internet, this is a resource that came from the Internet, and we can find some sort of standard geometries and use the information for those geometries. In this case, you can see for a hollow rectangular cross-section, if we have a width labeled B and a depth labeled D with an inside and an outside value, there is a formula here expressed for the area moments of inertia, both vertically, or both around the horizontal axis Ixx or around the vertical axis Iyy. So we're going to go ahead and write those formulas down on our page here. So here's the formula for the moment of inertia around the x-axis. It's equal to the B outside D outside cubed over 12 minus B inside D inside cubed over 12. Notice this has a little bit to do with our sort of standard idea of area times D squared. D times D is effectively the area, and then you have two additional Ds involved distances, where our D is going to be the 6 inches and our B is going to be the 4 inches. Well, let's go ahead and plug in the appropriate numbers. Our outside width B is going to be our 4 inches. Let's go ahead and plug in the appropriate numbers. Our outside width B is going to be the 4 inches and we'll multiply that by our outside depth, which is the 6 inches cubed. And we divide that product by 12. Similarly, the inside breadth is going to be 5 inches, which is the 6 inches minus 2 thicknesses, a thickness on the top and a thickness on the bottom. And we're going to multiply that, actually it's the thickness on the right and the thickness on the left for our value B, which is going to be 3 inches, 4 minus each of those half inches, times the 5 inch value I was just talking about at the interior height cubed. And again, divide that product by 12. We might want to do a little bit of cancellation here if we want to simplify our values. Divide this by 4 here and we end up with a 3. And if we go ahead and do that multiplication, we end up with a value of 72 inches to the fourth minus 125 over 4 inches to the fourth. Or if we do the math there, 40.75 inches to the fourth. So that would be our value for the box beam applied around the central x-axis. If we instead wanted to apply it around the central y-axis from y to y, we would execute the same formula, but in this case we would reverse the B and the D. So our value for Iyy is going to be this time the 6 inches times the 4 inch value cubed minus the 5 inches times the 3 inch value cubed. Notice when we do the calculations there, we get 20.75 inches to the fourth. Compare that to the earlier version of Ixx, which was 40.75 inches to the fourth. You can see that if we stand the beam up, so it's taller in this length direction, that will offer more resistance to a bending moment. Now let's consider the example of a hollow cylinder. In this case the information for the hollow cylinder is that it has an outer diameter of 10 centimeters and a thickness of 1 centimeter. So again, we go and look at a chart of typical calculation values here. And we'll see a formula there for a hollow circular section. And that hollow circular section says that both of the values are going to be equal to each other. The Ixx and the Iyy are going to have the same value and that value is going to depend on the diameter, the inner and the outer diameter, where big D is the outer diameter and little D is the inner diameter. In this case the letter P represents the value pi. So taking that information back to our calculation, we can write this value that Ixx and Iyy are equal to pi divided by 64 times the outer diameter minus the inner diameter where each one is to the fourth power first. And notice this makes sense. It's going to give us a value in length to the fourth power. Well, now it's simply a matter of plugging in the appropriate numbers. Pi over 64 times 10 centimeters to the fourth power minus, in this particular case we said the thickness was 1 centimeter. Well, that means our inner diameter is going to be 8 centimeters to the fourth power. Pi over 64 times the value of 10,000. And now we also need to have a value for 8 to the fourth power, 4,096. And notice both of these together are going to be centimeters to the fourth. And so our final value here is going to be 289.8 or using the significant digits we have 290 centimeters to the fourth. And again, that applies to both Ixx and Iyy given the symmetry of the cylinder.