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how do I extrude the surface enclosed by the selected edge?

Answers
Use the Fill feature first, then Extrude the face of the filled surface. Or you could create a sketch and apply the Use or Intersect sketch feature, then Extrude the sketch region.
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How cool would it be you could just select some edges and THEN extrude revolve etc. adding in a plane axis or surface to complete the feature. I found myself trying to do the same as OP early in the learning process.