How does Volume work with integration? Using a cross section suppose, as described here: Area formula Paul Notes

Suppose this is: $y = f(x)$.  
He says the volume is:
$$\int_{a}^{b} A(x) dx$$ 
But how does area over that interval give the volume? I mean, 
You are not getting the area of a cross section (the YELLOW part) are you? You are just taking the area of The 2D region, which isnt the cross section?

Then how does summing that area give the volume when its not the cross section? The actual cross section is the yellow part. 

$A(x)$ represents the area of the cross section. It is a circle of radius $f(x)$. Thus $A(x)=\pi\,f(x)^2$.
To see that the integral gives the volume imagine the solid of revolution is a salami and you cut it in thin slices of width $dx$. The volume of each slice is $A(x)\,dx$, and when you sum them all you get the volume of the salami. 
