The Fundamental Theorem of Calculus

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The fundamental theorem of calculus establishes the relationship between the indefinite integrals and differentiation by use of the mean value theorem. It also states that the indefinite integrals and differentiation are inverse operations. Note the graphs below.

Mean Value Theorem for Integrals
If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that
   _b
§  f(x) dx = f(c)(b - a).
  ^a

Average Value of a Function
The value of f(c), given in the Mean Value Theorem for Integrals, is called the average value of f on the interval [a, b]. Definition: If f is integrable on the closed interval [a, b], then the average value of f on the interval is
                 _b
1/(b - a) §  f(x) dx.
                 ^a

The Fundamental Theorem of Calculus
If a function f is continuous on the closed interval [a, b] and F is anantiderivative of f on the interval [a, b], then
   _b
§  f(x) dx = F(b) - F(a).
   ^a

The Second Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then for every x in the interval,
           _x
d/dx[§  f(t) dt] = f(x).
           ^a