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


b

a 
f(t) dt = F(b)-F(a).