Suppose I have the following trapezoid: 
Now, I used two methods to calculate the x coordinate of its centroid:
- $x_c = \frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{0+0+3+4}{4} = 1.75$
The second method is the weighted method, it involves dividing the trapezoid into a triangle and rectangle, and using the formula:
$x_c = \frac{A_1 \cdot x_{c1} + A_2 \cdot x_{c2}}{A_1 + A_2}$
Where $x_{c1}$ is the x coordinate of the centroid of the rectangle, which is of course 1.5, and $x_{c2}$ is the x coordinate of the centroid of the triangle, which is: $x_{c2} = \frac{3+3+4}{3} = \frac{10}{3}$. $A_1$ is the area of the rectangle and it is $3$, and $A_2 = \frac{1}{2}$
Plugging this in, I get $$\frac{3 \cdot 1.5 + \frac{1}{2}\cdot \frac{10}{3}}{3+0.5} = 1.761904762$$
My question is: As much as these two values are close, they aren't identical. I thought that the first formula worked when both bases are parallel to the x-axis, so I naturally thought these two should have the same value. Could someone explain why this is not the case?
Edit: I am using the Center of Sums method for defuzzyfication, the description of which can be found here: https://cse.iitkgp.ac.in/~dsamanta/courses/archive/sca/Archives/Chapter%205%20Defuzzification%20Methods.pdf
Posting this Comment in Answer format because it is getting too long.
Original Issue :
You are mixing up Centroid & Center of gravity.
You can try making two triangles ABC ADC & calculate , then try making two triangles ABD & CBD & calculate. You will get more Different Answers.
Basically Centroid is average of x-coordinates , while Center of gravity uses weights.
Updated Issue :
It is neither Euclidean geometry nor analytic geometry nor Centroid : It is a defuzzyfication method , where the Answers calculated are not going to correspond to some geometric Center or Center of gravity or ETC.
Document says this about "Center of Sums (COS) Method" :
In this method, the overlapping area is counted twice.
Naturally , this is not going to correspond to Centroid.
Document says this about "Weighted Average Method" :
This method is valid for fuzzy sets with symmetrical output membership functions and produces results very close to the COA method.
Implicitly & Explicitly , it is saying that each Method will give Different Answer.
There are at least 8 Methods listed , with each giving Different Answer.
It is not valid to compare with Strict Precise Concepts like Centroid & Center of gravity.
It is "fuzzy" logic due to "fuzzy" nature of the calculations involved.
It is Expected that we will get Different Answers via Different Methods here !!
Settling the Overall Issue :
It is not true that Polygon Centroid is the Average $x$ value of the vertex Co-Ordinates.
Average $x$ Co-Ordinate is valid when we have finite number of Points.
Hence , when we take Polygon (trapezoid) Edges or Polygon (trapezoid) Area , we have infinite number of Points.
We can not use the Average of $x$ Co-Ordinate on those vertex Co-Ordinates.
Hence $1.75$ is wrong & $1.761904762$ is Correct.
In fuzzy logic , that is okay , because we want to use fast calculations to get Close Answers.
Proof that Averaging is wrong :
When we have triangle $WYZ$ , we know the Centroid $x$ Co-Ordinate , which is "Stationary".
Let us include a new Point on the Base , to make a Quadrilateral $WXYZ$ , which is Essentially the Same triangle.
We should Expect that Centroid is "Stationary" & will not change.
Yet , we are now adding $4$ numbers & Dividing by $4$.
More-over , $X$ can move along the Base , which will Contribute various $x$ Co-Ordinates to the Average , hence this way to calculate Centroid will not give Same Answer.
Where is it going wrong ? Answer : Averaging is valid when we have finite number of Points. Averaging is invalid when we have infinite number of Points.
Why is is still used in fuzzy logic : We want fast calculations to generate Close Answers , not Accurate Answers.