Fundamental Theorem of Calculus from "summary" of Differential Calculus by S Balachandra Rao
The Fundamental Theorem of Calculus is a powerful and elegant result that connects two seemingly unrelated branches of mathematics - differentiation and integration. It establishes a fundamental relationship between the two operations, showing that they are, in fact, inverse processes of each other. The theorem is divided into two parts, known as the First and Second Fundamental Theorem of Calculus. The First Fundamental Theorem states that if a function f(x) is continuous on a closed interval [a, b] and F(x) is any antiderivative of f(x), then the definite integral of f(x) from a to b is equal to the difference of F(x) evaluated at b and a, i. e., ∫[a, b] f(x) dx = F(b) - F(a). This result essentially tells us that the process of finding the area under the curve of a function by integration can be reversed by differentiation. In other words, if we know the antiderivative of a function, we can easily determine the definite integral over any interval. The Second Fundamental Theorem of Calculus complements the First Theorem by providing a method to find antiderivatives. It states that if F(x) is any antiderivative of a continuous function f(x), then the derivative of the definite integral of f(x) from a to x is equal to f(x), i. e., d/dx ∫[a, x] f(t) dt = f(x). This theorem implies that differentiation "undoes" integration, allowing us to recover the original function from its integral. In practice, the Fundamental Theorem of Calculus is a valuable tool for solving a wide range of problems in calculus, especially in evaluating definite integrals and finding antiderivatives.- The Fundamental Theorem of Calculus serves as a cornerstone of calculus, providing a deep insight into the relationship between differentiation and integration. It highlights the unity and interconnectedness of mathematical concepts, showcasing the beauty and elegance of mathematical theory.
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