It's always a sort of guessing game unless you have the equipment to test inverter boards/screens, but with your description of the problem I am going to guess it is an inverter problem. A bad screen will usually produce some sort of backlight or screwed up picture, but complete black sounds like the inverter. If you plan on replacing this yourself, you are going to have to open up the screen anyway, so be sure to look at the LCD cable on the back of the screen to check for any scorch marks, as I have had this happen in a few laptops, and it is a pretty cheap thing to fix.
The LCD cable connects to the back of the LCD, then to the inverter, and then to the graphics card. So yes, there is a shared cable.