STATEMENT OF PROBLEM

We consider the problem of manufacturing a curved panel. The panel we wish to make is the front, upper right-hand, quarter of the section of the cylinder x^2 + z^2 = 1 which is cut away by the cylinder y^2 + z^2 = 1. This panel is depicted below in outline on the left to right cylinder and within the open cylinder facing the viewer.

For the purpose of manufacturing sheet metal the sheet metal is flat. We must determine the equation of a function y = f(x) so that when we fold the piece around the left-to right cylinder of radius it will conform exactly to our desired piece depicted above. The following diagram shows the outline of the desired rolled piece PMB and the flat piece which must be cut out (PMC) in order to get the desired rolled piece when the cut piece is folded along the cylinder x^2 + z^2 = 1.

Your job is to find and verify the correctness of the equation y = f(x) which will do the job.

You are also to determine separately the precise area of cut out section PMC and the surface area of rolled panel PMB and confirm they are equal.