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:  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?

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 $f_1=f(x_1), f_2=f(x_2),\dots, f_n=f(x_n)$:

![enter image description here](

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.