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. 

It is as simple as 
\begin{align}
V 
&= 
\int\limits_V dV 
\quad 
\mbox{(adding up tiny Volume elements over the volume)} \\ 
&= 
\int\limits_a^b A(x) \, dx 
\quad 
\mbox{(adding up slices $dV = A(x)\,dx$ along the $x$-axis)}\\ 
&= 
\int\limits_a^b \int\limits_0^{f(x)} \int\limits_0^{2\pi} y\, dx \, dy \, d\varphi \\
\end{align}
Where
$$
A(x) 
= 
\int\limits_0^{f(x)} dA 
= 
\int\limits_0^{f(x)} 2\pi y dy = \pi f(x)^2
$$ 
results from a summation of circular strips $dA$
and the circular strips result from summing up the elements
$y\,dy\,d\varphi$ over the circle
$$
dA = \int\limits_0^{2\pi} y \, dy \, d\phi = 2\pi y\, dy
$$
