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Key Concepts

Antiderivative

An antiderivative of a function f is another function F whose derivative is f. Finding an antiderivative means determining a function F such that F'(x) = f(x). This concept is central since it allows us to relate differentiation to integration by identifying a function that 'undoes' the derivative of f.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the processes of differentiation and integration. It states that if f is continuous on an interval [a, b] and F is an antiderivative of f, then the definite integral of f over [a, b] is given by F(b) - F(a). This formula provides a practical method for computing the net area under f by evaluating its antiderivative at the endpoints.

Definite Integral

A definite integral represents the accumulation of a quantity, often interpreted as the net area between a function's graph and the x-axis over a closed interval [a, b]. Using the Fundamental Theorem of Calculus, the computation of a definite integral shifts from summing infinitesimal contributions (the limit of Riemann sums) to a simpler evaluation of an antiderivative at the bounds of the interval.

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