For g(x) = ∫ a to x f(t)dt, I would like to know why there is an x at the upper limit and why it not equal to b, what is the difference?

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- May 12th 2010, 04:21 AMSyNtHeSiSFundamental Theorem of Calculus, Part I
For g(x) = ∫ a to x f(t)dt, I would like to know why there is an x at the upper limit and why it not equal to b, what is the difference?

- May 12th 2010, 04:50 AMHallsofIvy
There's a very important difference- is a

**number**, a constant. is a function of x.

For example if f(t)= 2t+ 1, then while .

More subtle is the difference between and the , the "indefinite integral" or "anti-derivative". where we need the constant C since for any C. The "a" in determines the specific C: while . - May 12th 2010, 05:32 PMSyNtHeSiS
Makes sense. So basically it differs from an indefinite integral by not containing the constant c?

- May 13th 2010, 04:35 AMHallsofIvy