What was found
NORMAL FLAPS 20 TAKEOFF WITH REDUCED THRUST. AT 1000 AGL, NR 3 FIRE WARNING ILLUMINATED AND FIRE BELL SOUNDED. SHUTDOWN ENGINE PER QRH PROCEDURES. REQUESTED AND RECEIVED VECTORS TO DUMP FUEL OVER THE WATER, DUMPED DOWN TO MAKE 630 LANDING WEIGHT. AFTER ENGINE WAS SHUTDOWN, FIRE INDICATIONS IMMEDIATELY WENT OUT. A NORMAL LANDING APPROACH WAS ACCOMPLISHED. UPON INSPECTION, FOUND NR 3 ENGINE WITH CRACKED CASE. REMOVED AND REPLACED NR 3 ENGINE ESN 455289 (OFF).