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The Fundamental Theorem of Calculus 🧮🐸

The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration, two core concepts in calculus. It is divided into two parts:

1. The First Part states that if a function is continuous over the interval \([a, b]\) and \(F\) is the indefinite integral of \(f\) over \([a, b]\), then: \( F'(x) = f(x) \) for all \( x \) in \((a, b)\).
2. The Second Part asserts that if \( f \) is continuous over \([a, b]\), then the integral of \(f\) from \(a\) to \(b\) is: \( \int_a^b f(x) \, dx = F(b) - F(a) \) where \(F\) is any antiderivative of \(f\).

This theorem powerfully combines the concept of the derivative, representing rates of change, with the concept of the integral, representing area under a curve.

Explore more about calculus by visiting our detailed sections on Derivatives and Integrals.