B737 FLT CREW REPORTS OIL QUANTITY DECREASING ENROUTE WITH EVENTUAL LOSS OF OIL PRESSURE. ENGINE IS SHUT DOWN AND EMERGENCY DECLARED WITH LANDING AT DESTINATION ARPT.

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AT FL340; I NOTICED THAT THE OIL QUANTITY ON #2 ENG WAS FALLING OR GOING TOWARDS ZERO. BEFORE PUSH THE QUANTITY WAS AT 94-95%. OIL TEMP WAS NORMAL AND OIL PRESSURE WAS NORMAL. WE HAD A DEADHEADING PLT LOOK AT #2 ENG TO SEE IF THERE WAS ANY OIL STREAKS OR SIGNS OF AN OIL LEAK. THEY RPTED THAT THERE WAS NOTHING OUT OF THE NORMAL. THE CAPT AND I KEPT AN EYE ON THE ENG INSTS FOR A WHILE NOTING NO CHANGES OUT OF THE NORM; THEN SENT A MAINT MESSAGE STATING OIL QUANTITY DISCREPANCY FROM PUSHBACK; ALL ALONG THINKING OF A GAUGE ERROR. LATER IN THE FLT; NOTICED THE OIL PRESSURE SLOWLY STARTED TO FALL. THE CAPT AND I REVIEWED THE CHKLIST FOR LOW OIL QUANTITY AND WE EVENTUALLY DETERMINED THAT WHEN THE PRESSURE FELL BELOW THE RED LINE WE WOULD HAVE TO SHUT DOWN THE ENG. CALLED MAINT AND DISPATCH TO NOTIFY OF OUR SITUATION. PROCEEDED TO ZZZ THEN THE OIL PRESSURE FELL TO THE SHUTDOWN LIMITS. DID ALL APPROPRIATE CHKLISTS AND ACCOMPLISHED THE INFLT ENG SHUTDOWN CHKLIST AND DECLARED AN EMER WITH APCH. CONTINUED TO ARPT ENVIRONMENT AND THEN HANDED OFF TO ANOTHER CTLR. PROCEEDED ON A 170 DEG HDG DOWNWIND FOR RWY. WAS NOT BEING VECTORED FOR EXPEDITIOUS HANDLING. WE QUERIED THE CTLR IF AWARE WE WERE EMER ACFT AND DID HAVE AN ENG SHUT DOWN. HE THEN VECTORED SOMEONE OFF OF FINAL TO PUT US IN SEQUENCE THEN TURNED US ONTO FINAL THEN CLRED FOR THE APCH. TOLD TO SWITCH TO TWR; DID THEN FINAL MONITOR CANCELED OUR APCH CLRNC DUE TO AN F16 AT ZZZ AFB AND HAD TO WAIT TO CLR OF THE CONFLICT THEN RECLRED FOR THE ILS. APCH AND LNDG WERE JUST LIKE THE WAY WE DO THEM IN THE SIMULATOR AND UNEVENTFUL. HAD THE FIRE DEPT CHK THE #2 ENG; NO FIRES NOTED AND THEN TAXIED TO THE GATE UNEVENTFUL.