Stage 1: The elliptic paraboloid is parameterized as

The "back side" of this surface is plotted for      ( s-resolution is 16) and   ( t-resolution is 16) and is shown to the right. 

Stage 2: The cylinder is parameterized as

and is plotted for   ( s-resolution is 1) and    ( t-resolution is 32).  This portion of the cylinder is plotted without erasing the portion of the elliptic paraboloid already plotted and is shown to the right.

Stage 3:  It is now time to plot the "front-side" of the elliptic paraboloid, using the same parameterization as in stage 1, but plotting     ( s-resolution is 16) and    ( t-resolution is 16).  Set the "Plot Axes" option to "Off" and plot this portion of the elliptic paraboloid without erasing the previous plot to give the figure to the right.

Stage 4:  All that remains to be plotted is the portion of the cylinder that lies below the elliptic paraboloid.  First of all, note that the two surfaces intersect when   , or in terms of the parameterization for the cylinder, when   .  Modify the parameterization for the cylinder to

and plot without erasing the previous graph for   (s-resolution is 1) and   ( t-resolution is 16). The resulting graph is shown to the right.