While there isn't a specific name that we know of, "Undermounted Shower/Tub Combo" is what we'd call it in a pinch. The pricing for something of this configuration would vary on lots of factors, but in some cases a free-standing tub can actually be more expensive than a drop-in tub since the side profiles are not visible. For the Cohen Residence, we used an undermounted tub deck surrounded by custom marble stonework to achieve a high end profile.

Thanks. Ours is nearly complete and I guess I would call it, "Mini-undermounted shower/tub combo...low budget division!" :-)