Pääteoreema

Pääteoreema, also known as the Fundamental Theorem of Calculus, is a cornerstone of calculus that establishes a relationship between differentiation and integration. It consists of two parts: the first part states that if a function f is continuous on the closed interval [a, b], and F is the integral of f from a to x, then the derivative of F with respect to x is f(x). This part is often referred to as the Fundamental Theorem of Calculus, Part 1. The second part, known as the Fundamental Theorem of Calculus, Part 2, states that if f is continuous on the closed interval [a, b], and F is any antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a). This theorem is fundamental to the study of calculus and has numerous applications in mathematics, physics, and engineering. It allows for the evaluation of definite integrals using antiderivatives, and it provides a method for finding the derivative of an integral. The Fundamental Theorem of Calculus is a powerful tool that connects the two main branches of calculus: differentiation and integration.