What is Integral Quadrature? I'm reading this book:
Differential Quadrature and Its Application in Engineering
trying to find out what is "Integral Quadrature". here is a snapshot of the page which is about Integral Quadrature:

My questions is what are w(weighting coefficiont) and f(functional value) and could you please show me them on the figure 1.1?
It's so close to the concept of "Integration" which we learnt at school but I don't know why it make me crazy. this is what we learnt about integrating at school:
Introduction to Integration
P.S. I don't have enough reputation to add these tages: "Integral Quadrature" and "Differential Quadrature" would you please do it for me?
 A: I changed a little of one of the pictures in your link so you see those $x_1, x_2, \dots, x_n$ and their corresponding $y$-values $f_1=f(x_1), f_2=f(x_2),\dots, f_n=f(x_n)$:

Suppose the original interval is $[a,b]$, that is , you want to find the area under the curve $f(x)$ in the interval $[a,b].$ Now if $w_1=(b-a)/(n-1), w_2=(b-a)/(n-1), \dots, w_{n-1}=(b-a)/(n-1), w_n=0$, we get
$$\frac{b-a}{n-1}(f_1+\cdots+f_{n-1}),$$
which is the sum of the area of the rectangles using the height of the left boundaries. Notice that $(b-a)/(n-1)$ is the width of each rectangle.
Otherwise, if $w_1=0, w_2=(b-a)/(n-1), \dots, w_{n-1}=(b-a)/(n-1), w_n=(b-a)/(n-1)$, we get
$$\frac{b-a}{n-1}(f_2+\cdots+f_{n}),$$
which is the sum of the area of the rectangles using the height of the right boundaries. 
The integration quadrature is a generalization of this idea, with different weights on different $y$-values.
A: Maybe a good thing to think about is why you would want to weight the areas rather than just use a simpler method like the trapezium rule where you simply average all the trapeziums. For instance, if you have a very 'wavy' function you might want to use non-equally spaced points and weight them accordingly.
