There is no magical end of a projected image at a "black circle line" on your diagram it actually continues - This is where vignetting (darkening of the image at the edges) comes into play. The best image area on DX lenses, is the size of the DX sensors. It does not mean that it can not be used on FX or the image does not cover a FX image sensor. On Zoom lenses as Elvishefer pointed out, as you zoom the image circle become larger and covers more area.
The 35mm 1.8 does work with a FX sensor but that does not mean all DX lenses will work as well. I have seen sample photos of the 85mm VR macro on FX and that looked pretty nasty. Same goes for the 18-55mm lens. How well DX lenses work on FX sensors is lens by lens.
Thanks TTJ, I think I'm understanding this a little better now :)