There are two objects in this image.  There is a square and a cylinder (that's been boolean'd into pieces.)  What I want is to merge these two objects.  I know about ^J, the merge option under Mesh, and about boolean union.  What I don't know is which of these will join the meshes together so that there is literally no gap between the cylinder and square. I get the idea that join makes many objects into one. But I see nothing that shows the meshes are merged.  Same thing for union.  What I am looking for is one solid object.  Does this make sense? 

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/lG5MH.png