My brother was sweeping the house the other day, and I asked him:
Have you already swept the place under the bed?
As a learner I didn't find any other alternative to talk about the place / space under the bed.
How would native speakers say this?
I would say it's not wrong, the place is describing a location that, in this context, was identified by its relation with the bed. However, it isn't seem to bring significant meaning when added in the sentence.
Have you already swept under the bed?
Have you already swept the floor under the bed?
could express the same idea and, even to me, it sounds better in the ears