Using a cross section suppose, as described here: [Area formula Paul Notes](http://tutorial.math.lamar.edu/Classes/CalcI/Area_Volume_Formulas.aspx)

![Ruby Fapshire](https://i.stack.imgur.com/Eusiw.gif)

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?

![ruby gagshire](https://i.stack.imgur.com/t1DIw.gif)

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