Consider a surface . Let the orthogonal projection on XY plane of its portion S’ be the surface S. Divide S into elementary rectangles of area by lines parallel to x and y axis. Now consider prism have its base as these triangles and its length parallel to OZ.
The volume of this prism under consideration is bounded by surface S and surface is .
Hence the volume of the solid cylinder having S as base, bounded by the given surfaces with generator parallel to Z-axis is
Where integration is to carried over surface S.
In polar coordinates the above integration becomes