Fundamental Theorem Of Calculus

The Calculus is the most powerful tool the human mind has ever discovered.

In mathematical analysis and study, in engineering, science and practical life problems of  areas and tangents, rates of change,  differentiation and integration, probability, and measure theory

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The Fundamental Theorem Of Calculus * expresses the relationship between differentiation and integration namely that if the function F exists for which F as the integral of f also exists, where f is F' the derivative of F for which F'(x) = f(x) all x in [a.b] Then the integral of f over [a,b] equals F(b) – F(a)

In other words the function F is the anti-derivative of f

* Terms to be explained

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Given two points on a line the slope m or gradient, is change in y, over in x. For a straight line it is constant. In the limit as x (in extreme) changes infinitesimally the gradient of the tangent of a curve equals the derivative at x. A tangent line touches a curve and in one point as entering and leaving the curve in the same (single) point. 

The slope of a tangent equals the derivative at x. The integral F(b) – F(a) is the area under the curve f between x=a and x=b.

At first it is surprising, and that it seems the two, the tangent and area would have nothing to do with each other.

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Traditionally the credit for first studying Calculus goes to Isaac Newton of England and Wilhelm Leibniz of Germany in the 17th Century.

Georg Riemann placed integration in a formal theoretical foundation in the 1800s with infinite sums and limits of real analysis' so-called epsilon-delta arguments and Riemann partitions of x-intervals.

Much larger classes of functions are Lebesgue integrable, from beginning of 1900s, also a formal abstract theory of measure and probabilities, in breakthrough measure theory y-intervals are partitioned instead of x-axis.