Good day. Today we're going to talk about areas for two-dimensional bodies or lengths for let's say a wire or volumes for three-dimensional body. Also, we are going to introduce the concept of centroid. Now, the calculation of areas, lengths or volumes is important because of two reasons. On one hand, we know that gravitational loads are distributed loads. So if we want to calculate the total weight of a body, we need to sum up all the weights acting throughout the body, and clearly this depends on the volume of the body, or the length of the wire, or on the area of a plate whose weight we want to calculate. All right. So in addition, the calculation of areas will be shown to be very important whenever we have loads which are not constantly loads, but we have distributed loads with a sudden intensity. Again, in order to replace this distributed load with a single concentrated load we need to calculate the areas. The concept of centroid is important because once we have calculated the equivalent load, we need to also decide their position where this load should be placed in order to create an equivalent system. That means in order to create for example, the same moments of this equivalent load as the distributed loads produced. So let us start with a case of a two-dimensional object. Let's say this is an area A, and we have let's say coordinates x and y. We define the area A as the integral of all areas dA comprising this shape. So this areas could be represented in the Cartesian coordinates using this infinitesimal element, let's say dx and dy. So we could say that this is equal to double integral dx dy. Or if we were to use polar coordinates to represent different points within the shape. Let's say distance r and angle theta representing this point. Then the infinitesimal element we're going to consider is an element corresponding to an increment of the theta, and in this direction dr. So such little elements where this is dr, and this length here is rd theta. So we could write here that this is the double integral of such infinitesimal elements rdr theta. We could also have considered larger elements dA. For example, we could have taken little strips like this horizontal strips where this is dA. Before this was dA or this was dA. Now, in this case of this horizontal strip again, we're going to write now a one-dimensional integral dA of this infinitesimal strips. In the case where we would take for example vertical strips, again, we could have considered very classic dA and express this area as the sum, the integral of all these little areas dA as x now moves from the minimum distance to the maximum distance shown here. Now, once this area has been calculated using one or two-dimensional integrations, then the next question is, where is the center of this area, the geometric center of this area which we refer to a centroid? Where is the centroid C of this area located? Now C is an area the coordinates xc and yc of this centroid C, xc for example, if we ask what is the x that most representatively represents this area? It is clearly a number which is larger than this minimum distance from here to here, let's say this is x minimum, and it's a number which must be smaller than the maximum distance, let's say this is x maximum. So it's a number between x minimum or x maximum. Is it the average? Is x minimum plus x maximum over two? Clearly not. It depends really on the shape of this area. Now, for this area, for this little slip for example, all points clearly have the same x. If we take another strip here for example, all the points within this trip also have the same another but the same x. So the average x which we call xc is the weighted average of all these x distances, where this x for example representing a smaller area is weighted less than this x which represents a big area. So each x is weighted by the area it represents the A, and we normalize this by the total area A. That means this can be thought as integral of x times some waiting which is dA over A. We can think about this as an integral of X's, this x is weighted by the relative area dA over A. This denominator clearly is the total area A. Similarly, the average y distance is the weighted average of all the y distances given by this expression. Now, for vertical strips, the x is uniquely determined. There is one x representing the entire strip. In the case where we have a horizontal strip, the x representing such as trip would be the x which corresponds to the midpoint of this strip. So this would be the x when we consider this horizontal strip. When we consider an element using polar coordinates, then the x corresponding to this element would be r times cosine of theta. While the y should be r sine of theta. So we derived here the formulas that give the area, and the centroid distances for the case of a two-dimensional body. Notice that the centroid does not have to lie within the shape we are considering. For example, in the case of let's say a donuts, the case of a donut the centroid is here and it does not belong in the donut shape. Now, let us consider now two more cases. The case of a one-dimensional line L within again, some coordinate system xyz. We define the length of the line as the integral of little infinitesimal elements dL, and the coordinates of the centroid the center of this line let's say sensors here. The center of the line would be for example, the integral again a weighted average of all x's, x times the length dL it represents over the total length. Similarly for yc and zc. Lastly, in the case of a three-dimensional body. Again, the volume of this body would be the integral of infinitesimal volume elements dV, this infinitesimal volume elements for examining Cartesian coordinates would be little cubes dx dy dz, so it could be written as a three-dimensional integral of dx dy dz over the entire volume. Or we could have used different strips or slices to also use infinitesimal quantities of less order. But this instead of three-dimensional, we could use two-dimensional or first-order infinitesimal quantities. The central distance again, if the central of this element is somewhere let's say here, the central distance xc is again similar to before the average distance x, where x's are weighted according to the volumes they represent. Such a weighted average of x's in all the cases where the x's are weighted according to the relative area, relative length, or relative volume they present. Let us look at some example where we're going to calculate the area and centered of some simple but often encountered geometric shapes.
