Fundamentaltheoreemiksi

The Fundamental Theorem of Calculus (FTC) establishes a bridge between differentiation and integration for functions defined on an interval. It comprises two parts that connect the process of accumulation with the operation of differentiation.

First Part: If f is continuous on [a,b] and F defined by F(x) = ∫_a^x f(t) dt for x in [a,b], then F is differentiable on (a,b) and F'(x) = f(x). Thus the derivative of the accumulation function recovers the original function.

Second Part: If f is continuous on [a,b] and F is any antiderivative of f on [a,b] (that is, F' = f), then ∫_a^b f(x) dx = F(b) − F(a). This allows evaluation of definite integrals via antiderivatives.

History and significance: Independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, the theorem provides a rigorous justification for procedures used in solving problems involving area, accumulation, and physical quantities. It underpins much of modern analysis and its applications in science and engineering.