Alright ,so I went to several stores and checked out laptops with Ati Radeon 3200/4100/4200 graphic cards. I know all of the above are integrated GC's. But on some machines, Win 7 showed 256MB Ram of dedicated memory while on some, it didnt. Can integrated cards have dedicated memory, if yes, does it make them a little more efficiency
They can have memory on occasion, but it really doesn't make a difference. The chip itself is so weak that you're probably not going to be able to even fill 256MB of memory with the lowered settings you'll have to use.
The ones with dedicated memory would always be better, though in this case it is just a teeny-bit better. Since the videos card you mentioned are already weak themselves you'd probably gain 5% or less with the dedicated memory.
Though I'd recommend to always get the ones with dedicated memory, they'd be a teeny bit faster and they'd let you use your whole memory instead.
Characterizing modern integrated graphics chips as "weak" isn't fair, IMHO unless you are comparing them to high end chips and then everything else is weak. I recently built a box with ATI 4200 integrated graphics and decided to try it out before adding a discrete card. I'm glad I did because it runs everything more than a couple years old, for example, Half Life 2 runs as well as it ever ran on my most powerful machine. I recommend a little experimenting before spending more than necessary.