The Fundamental Theorem of Calculus provides a technique to evaluate a definite integral without relying on the Reinmann Sum technique. The theorem uses techniques associated with the indefinite integral (or anti-derivative) to solve problems involving definite integrals.
The theorem states:
For a function f(x) continuous on an interval [a,b], and assuming any antiderivative (indefinite integral) along that function is represented as F(x), then its evaluation is the difference of the anti-derivative evaluated at the upper limit, and the anti-derivatve evaluated at the lower limit. Or:
${\int }_{a}^{b}\phantom{\rule{3px}{0ex}}f\left(x\right)\phantom{\rule{5px}{0ex}}dx=F\left(b\right)-F\left(a\right)=F\left(x\right){\right]}_{a}^{b}$
Note that the constants of integration ('C') that had always been added for indefinite integrals are the same for the bounds of integration (F(a) and F(b)), so the constants are ignored since they would subtract each other out and yield 0.
In this first example, we evaluate:
${\int }_{2}^{4}{x}^{2}\phantom{\rule{5px}{0ex}}dx$
Fundamental Theorem of Calculus