Fundamental Theorem of Calculus
We are all used to evaluating definite integrals without giving the
reason for the procedure much thought. The definite integral is
defined, however, not by our regular procedure but rather as a limit of
Riemann sums. We often view the definite integral of a function as the
area under the graph of the function between two limits. It is not
intuitively clear, then, why we proceed as we do in computing definite
integrals. The Fundamental Theorem of Calculus justifies our
procedure of evaluating an antiderivative at the upper and lower
limits of integration and taking the difference.
Fundamental Theorem of Calculus
Let f be continuous on [a,b]. If F is any antiderivative for
f on [a,b], then
ó õ |
b
a
|
f(t) dt = F(b)-F(a). |