PA-24 pilot reported the engine stopped during approach due to carburetor icing as a result of the pilot not applying carburetor heat. Pilot landed safely.

Narrative

It was a quiet day at the untowered airport due to heavy clouds and light rain most of the day which burned off around XA:30. Weather had been IFR for most of the last 10 days with another extended round of IFR forecast to roll in soon; so despite being late in the day (sunset around XB:30) I decided to go up for a short local flight to warm up the oil and charge the battery. On my return I had just entered the downwind for Runway XX when another pilot called up looking to depart on XY - weather was calm; we were the only 2 planes in the area; and with daylight starting to fade it made sense to me that he'd want to depart from closer to parking vs taxiing the length of the runway. I acknowledged his request; we coordinated that I'd extend my downwind and turn base under and behind him as he departed straight out; and I monitored his takeoff and turned in as coordinated. The extended downwind left me a touch high and fast on the obstacle avoidance dogleg base / final so on short final I pulled back to idle; dropped full flaps and slipped in on an intentionally long landing due to the higher then normal approach speed. The remainder of the landing seemed uneventful until on rollout I noticed the prop wasn't spinning. I coasted off the runway at the next turnoff and once it came to a stop on the taxiway I got out and pulled the plane over to a nearby ramp. I opened the cowls and on inspection I didn't see anything obviously wrong; but while conducting the inspection I noticed water dripping from the induction drain. I checked in the airplane and the carb ice knob was in the closed position. I was able to restart the engine without issue and did a successful runup and mag check. In hindsight I realized that the opposite direction takeoff call had come in while I was doing my landing configuration checklist and adding the task of tracking their takeoff roll and departure had shifted my attention and I did not complete my configuration checks. Additionally the abnormal pattern due to extending the downwind shifted my attention away from my normal configuration checks on base. As a result I missed opening carb heat and experienced a carb ice induced uncommanded shutdown on short final or rollout. Once stopped on the ground for a few minutes residual engine heat melted the ice and allowed me to restart and verify normal engine operation.