On-line Math 21

5.2  The Fundamental Theorem of Calculus

Let f be a continuous function on an open interval containing [a,b] . Let F(x) be any antiderivative of f . Then:

ó õ b a  f(x)dx = F(b)-F(a): = F(x)| ba.

Proof:

Set G(x) as before to be

G(x): = ó õ x a  f(t)dt.

Then, by the First FTC, G¢(x) = f(x) . On the other hand, we assume that F¢(x) = f(x) . A corollary to the MVF says that two functions with the same derivative can only differ by a constant, so F(x)+C = G(x) . Thus,

ó õ b a  f(x)dx = G(b) = G(b)-G(a)  (since G(a) = 0) = (F(b)+C)-(F(a)+C) = F(b)-F(a) = F(x)| ba.

Copyright (c) 2000 by David L. Johnson.