Thanks all.
I don’t think it’s a cooling issue, but what it is…well, the story so far:
Spent two hours on phone to Apple Care (and I hate to think what that will cost!).
They told me I had several problems, two related to internet, two to the computer.
First, they said I had corrupted caches or pref panes, which they guided me to delete. Then they said that four GB of RAM was not enough to support basic word processing, safari and login items. I turned all the login items off.
So this morning when I turned it on it seemed ok, although I still felt it lagged a little. This evening, when I turned it on, it told me there was no airport card installed. I thought if I rebooted it might help, so I turned it off. When I tried to turn it on again, nothing happened. It seemed to be off, but was emitting a buzzing noise, and it got very hot, even though it was on a solid surface with plenty of ventilation. It eventually stopped doing this after a couple of hours.
Now it will not turn on at all 
Before calling Apple, I had used TechTool Deluxe to test everything and it came back clear. I guess I’ll have to call Apple again.
But I was/am curious about the idea of 4GB of RAM being too little. So I ran Activity Monitor on the family MBP which also has 4GB. Although there were more processes running on that than on the MBA, and many were the same ones, the RAM usage was far lower. And although a lot of the RAM was in use, there was no lagging or mis-registering of my key-strokes.
I can’t directly compare the logs since I can’t boot the MBA, but as an example:
Single Safari window on Yahoo home page:
MBP: c.140 mb
MBA: c.350 mb
The advisors I spoke to (I had to speak to two) both told me the MBA levels were normal, but is that the case? I don’t see why there should be such a difference!
So maybe it is the logic board?
Anyway, I hate to ask another favour, but is anyone willing to give me an insight into their Activity Monitor figures? And is 4GB (or less) enough for what you do?
Apologies, but with another paid call to make, I’m questing for some extra know-how. Although I do have some access to another computer, all my University work is on the MBA. I have a Time Machine back up, and will only lose one night of work if the MBA has failed altogether, but I can’t easily restore everything I need to another machine, so I really, really need a quick fix if possible 
Thanks,
H.