FWIW, when you notch a joist, you effectively make the whole joist the equivalent of the thinnest section. The top and bottom are the parts that provide the most strength, either in compression on the top or tension on the bottom. When there's a gap from a notch, the bending moment's location changes and it gets much weaker. That's why there are rules about where it is acceptable to put holes in them (basically, avoid the ends, and limit their diameter). I'd probably box it out, but see what the others have to say.