Defuzzification using centroid and weighted average

Question

I was reading the book Fuzzy Logic by Timothy. J .Ross I don't understand how this defuzzification result was obtained. This is the example in the book:

The centroid method equation is :

I don't understand how the equations for $\mu_B(Z)$ was obtained. That is I don't understand for example, how from 3.6-4 it was integrated as $(z-3.0)\over 2$

Also in the weighted average method ,

I don't understand how the centroid values of respective shapes are obtained.
Can someone please help me to understand this?

Answer 1

The values are read from the graph, and straight-looking parts are interpreted as straight lines, meaning: assumed to be of the form $kx +b$ in the corresponding interval. Then the $k$ and $b$ are solved to find equation for the line. If $z = 3.6$ then $\frac{z-3}{2} = 0.3$ for example and at $z = 4$, it's $\frac{z-3}{2} = 0.5$.

Answer 2

Calculate the equation of line. as y=mx+c and m=(y2-y1)/(x2-x1) Here y2=.5 ,y1=.3, x2=4, x1=3.6 so m = .2/.4 = .5 . So y=.5x+c and c=-1.5 so y=.5x-1.5 which is equal (x-3)/2.