Overheard at a local McDonald's:

Sam:   I was wondering if I should call this guy a definite or an indefinite integral: f(t) dt.
Sally:   Is "x" a constant?
Sam:   Uh ... no, it's a variable.

Sally:   Then it's definitely an indefinite integral.
Sam:   OOPS! Sorry, x is a constant.

Sally:   Then it's definitely a definite integral.
Sam:   Well, if v(t) is the velocity, then isn't v(t) dt the distance travelled at time T?
Sally:   Obviously! I assume that the time, T, is constant.
Sam:   I've always had trouble making time stand still. How about f(t) dt ?

Sally:   I assume from your notation that "b" is a constant, so it's definitely a definite integral.
Sam:   Can I change my mind? I think I may want to vary "b".

Sally:   If you want to vary it, then you must never give it a name at the beginning of the alphabet!
Sam:   What about f(t) dt ?
Sally:   Do you want to vary "r"?
Sam:   Maybe I do and maybe I don't.

Sally:   Then I can't answer your question until you've made up your mind!

Sam:   Well, until I decide, can we give it some name ... like maybe an indefinite definite integral? Sally:   You can call it "Snoopy" if you like, but it'd be your private nomenclature. Sam:   Well ... uh ... doesn't the definite integral have a precise definition? Sally:   Of course! You subdivide the interval [a, b] into n subintervals, pick a point in each subinterval at which to evaluate the function, then you sum ... Sam:   Wait! If I choose the wrong name for the end points of the interval ... does it become an indefinite integral? Sally:   Of course! Names given to variables and constants are important. Things change when their name changes. Haven't you learned anything? Sam:   Uh ... you're saying that, if I want to vary "b", then a definite integral becomes ... becomes ... Waitress:   Would you like fries with that?

See: Math Stuff