Revolving a plane figure about an axis generates a volume. Consider the region between the graph of a continuous function  y = f(x) and the x-axis from  x =  a to x =  b.

This animation demonstrates how a surface area is generated by revolution and then how the sum of disks results in a volume.    

                                             
but the surface area is more complicated.

The function need not be one of the standard plane figures found in elementary geometry.  The perpendicular cross-section, or slice, is still a circle.

Source:curvebank.calstatela.edu