A method of cooling an electric induction furnace for melting and holding a reactive metal or alloy is provided where the electric induction furnace has an upper furnace vessel and an induction coil in a modular inductor furnace is positioned below the upper furnace vessel with a melt-containing vessel positioned inside the induction coil with a gap between the outside surface of the melt-containing vessel and the inside surface of the induction coil that is used to circulate a cooling fluid for cooling the melt-containing vessel to inhibit leakage of the reactive metal or alloy melt from the melt-containing vessel. The melt-containing vessel can be integrated with a cooling system for cooling the melt-containing vessel. Modularity of the melt-containing vessel, induction coil and cooling system facilitates servicing of the modular inductor furnace without disassembly of the entire electric induction furnace.