thanx you guys. I know how to do those with my eyes closed, its just that my AP book has a real habit of doing things w/o thouroughly explaining them. thanx again ill tell my friends about this site!
2nd FTC: If f is any riemann integrable function on the closed bounded interval [a,b], and G is a Lipschitz - continuous function such that for every point x where f is continuous, G is diiferentiable at x with G'(x) = f(x), then the integral of f from a to b, equals G(b)-G(a).
Recall that G is lipschitz continuous on [a,b] if there exists a constant K such that for all points u,v in [a,b] we have |G(v) - G(u)| <= K|v-u|.
in most books it says that if f is continuous on [a,b] and G(x) is the integral of f from a to x, then G is differentiable on [a,b] and G'(x) = f(x) for every x in [a,b].
The more general statement is that if f is a Riemann integrable function on [a,b] and G(x) again is the injtegral of f from a to x, then G is Lipschitz continuous, and G is differentiable with G'(x) = f(x) at those points where f is continuous.
Then to derive the 2nd thm from the first you need the generalized mean value theorem, that a function G which is lipschitz continuous and has derivative equal to zero almost everywhere (i.e. except on a set of measure zero) is constant.
in most books the 2nd thm just says that if f is continuous on [a,b] and G is continuous on [a,b] with G'(x) = f(x) for all x in (a,b), then the integral of f from a to b, equals G(b)-G(a).