In this video I will use one example to demonstrate how to determine the area moment of inertia of composite areas. 
First let me quickly go over the method we are going to use. 
If you recall, I mentioned the similarity between area and area moment of inertia. 
For example, as a physical quantity they are both always positive.
And if you wish to find the total area of this composite shape, all you need to do is to add the component areas together. 
It’s the same for area moment of inertia. 
Of course there’s a difference because unlike area, moment of inertia must be calculated about a specified axis. So let’s say 
if you want to find the moment of inertia of this composite shape about the x axis, you simply need to add up 
the moment of inertia of each component shape about the same x axis.
But how do we find the moment of inertia of the component shapes about this axis? 
We can apply the parallel axis theorem. 
For that we need to find the moment of inertia of each component shape about its own centroidal axis.
And as I said before,
this information is common and can be found either online, or from your engineering textbook or handbooks. 
So let’s look at this example. We need to find the moment of inertia of this composite area about the centroidal axis of the composite area. 
This could be a very typical example in the mechanics of materials class.
This could be the cross sectional area of a composite beam, and in order to do the stress analysis, we need to know the moment of 
inertia of the cross sectional area about a so-called neutral axis, which in this case is the horizontal centroidal axis. 
But the problem is, we don’t know where exactly it is. Therefore we need to first apply what we have learned before,  
and find the centroid location of this composite area first. 
So the very first step we do is to define our component areas. Sometimes there are multiple ways to do it but for this example, it is quite 
obvious that we want to choose these 4 rectangles to be the component areas. 
If you recall what I said before about finding centroid location, 
location is always described in relation to some reference. So the 2nd step is to choose a reference line.
Again, it is up to you how to choose this reference line, and as long as you are consistent
and accurate in your calculation, they all should point to the same location for the centroid. 
I always just choose the bottom of this shape to be the reference line. 
This is where the vertical coordinate y equals to zero. 
Then step 3, 
determine the centroid location of the composite area using the formula we learned before.
As you can see, you need to know the area of each component area as well as the total area.
And you also need to know y tilde, the location of the centroids for each component area.
Since they are all rectangles, the individual centroid location shouldn’t be difficult to find.
Just make sure y tildes are all expressed with respect to the same reference line.
Plug the information into the equation 
and get y bar
which is to 202.7 mm. That is the centroid location for the composite area. 
And we can mark the centroidal axis, the x axis, on this composite area.
And see that I kept all the information of the areas on this slide because we will still need that information later.
Now step 4, 
determine the area moment of inertia of each component area about the x axis.
To do that, we need to use the parallel axis theorem
Parallel axis theorem states that if we know the moment of inertia of an area about its own centroidal axis, then we can use it to calculate 
its moment of inertia about any parallel axis by adding the term A d squared to it. 
A is its area and d is the perpendicular distance between the two parallel axes.
So let’s start with component area 1. 
Note that due to symmetry, component area 1 and 2 are the same. 
So for component area 1, which is a rectangle, the moment of inertia about its own centriodal 
axis is calculated as 1/12 times b h to the 3rd power. 
b is the with of this rectangle which is 20 mm and h is its height which is 100 mm,
and we get 1.667 times 10 to the 6th power in the unit of millimeter to the 4th power. 
Remember what I said about the area moment of inertia, it always has the unit of length to the fourth power.
And because we know the location of its centroidal axis, 
and we also know the location of the centroidal axis of this entire composite area,
therefore the perpendicular distance between these two parallel axes is d_1 equals to 27.3 mm.
Therefore, according to the parallel axis theorem 
we can calculate the moment of inertia
of component area 1 as well as component area 2 with respect to the central axis,
the x axis, is 3.158 times 10 to the 6th power millimeter to the 4th power.
And we do the same thing for component area 3.
And then the same thing again for component area 4.
And lastly we simply add them all together, and get 
the moment of inertia of the composite area about the x axis, which is also the centroidal axis of this composite area. 
And that's the answer to this problem.
Now normally this is unusual but just for practice purpose, what if we are also asked to find the moment of inertia 
of this same composite area about the x” axis? 
Since we already know I_x, which is the moment of inertia about the x axis,
we also know the total area of this composite area
and we also know the perpendicular distance between these two parallel axes,
we can simply apply the parallel axis theorem to this entire composite area
and get the moment of inertia about the x'' axis.
As you can see it is a lot larger then I_x. As I mentioned before
for all the parallel axes
the moment of inertia calculated about the centroidal axis is always the smallest one.
