"Black borders" can mean a lot of different things? Are there borders around the entire video, or just the top/bottom and left/right?
YouTube has a zoom function that can be applied to zoom in the entire image. Their encoder may be auto-detecting the black space if there are borders around the entire video frame?
Any decent encoder can crop (same as zoom) X pixels form the tops and sides. Whether or not you want to use FFmpeg depends on the format you're encoding to. It does a sloppy job with some formats, like FLVs (sadly that's what most people use it for).
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