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 integrat...
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