The Attempt at a Solution
I know that given [itex]f(x)=c[/itex], the integral from [itex]a≤x≤b[/itex] is [itex]c(b-a)[/itex] (at least, I hope I know that! :D).
Is the integral the same value if you don't include an endpoint? That is, if you were evaluating [itex]f(x)[/itex] from [itex]a≤x<b[/itex]?
Intuitively I think it both integrals should be the same value, for 2 reasons:
1) The two integrals are the same except one of them doesn't include the ordinate of b (does not include [itex]f(b)[/itex] in the set of function values). This can be represented as a vertical line which would have no area.
2) I'm even less sure about this bit, but surely you could use a limit to show that evaluating the integral from [itex]a≤x<b[/itex] - "from a to as close to be as you can get" - is the same as evaluating it up to and including b?
Like I said, I don't really know, and these two reasons are just based on my gut feeling. I'd like to understand this properly, and answering this for me would be awesome, so thanks in advance! I equally want to know what the flaws there are in my thinking :)