The Fundamental Theorem of Calculus defines the relationship between the processes of differentiation and integration. That relationship is that differentiation and integration are inverse processes.
The Fundamental Theorem of Calculus : Part 1
If f is a continuous function on [a,b], then the function denoted by
is continuous on [a,b], differentiable on (a,b) and g'(x) = f(x).
If f(t) is continuous on [a,b], the function g(x) that's equal to the the area bounded by the u-axis and the function f(u) and the lines u=a and u=x will be continuous on [a,b] and differentiable on (a,b). Most importantly, when we differentiate the function g(x), we will find that it is equal to f(x). The graph to the right illustrates the function f(u) and the area g(x).
The Fundamental Theorem of Calculus : Part 2
If f is a continuous function on [a,b], then
where F is any antiderivative of f.
If f is continuous on [a,b], the definite integral with integrand f(x) and limits a and b is simply equal to the value of theantiderivative F(x) at b minus the value of F at a. This property allows us to easily solve definite integrals, if we can find the antiderivative function of the integrand.
Parts one and two of the Fundamental Theorem of Calculus can be combined and simplified into one theorem.
The Fundamental Theorem of Calculus
Let f be a continuous function on [a,b].
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