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# First Fundamental Theorem of Calculus

In the most commonly used convention (e.g., Apostol 1967, pp. 202-204), the first fundamental theorem of calculus, also termed "the fundamental theorem, part I" (e.g., Sisson and Szarvas 2016, p. 452) and "the fundmental theorem of the integral calculus" (e.g., Hardy 1958, p. 322) states that for a real-valued continuous function on an open interval and any number in , if is defined by the integral (antiderivative)

then

at each number in , where is the derivative of .

Unfortunately the terminology identifying he "first" and "second" fundamental theorems in sometimes transposed (e.g., Anton 1984), so care is needed identifying the meaning of these appellations when encountered in the wild.

Fundamental Theorems of Calculus, Second Fundamental Theorem of Calculus

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## References

Anton, H. "The Second Fundamental Theorem of Calculus." §5.10 in Calculus with Analytic Geometry, 2nd ed. New York: Wiley, pp. 345-349, 1984.Apostol, T. M. "The Derivative of an Indefinite Integral. The First Fundamental Theorem of Calculus." §5.1 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 202-204, 1967.Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cambridge, England: Cambridge University Press, 1958.Sisson, P. and Szarvas, T. Single Variable Calculus with Early Transcendentals. Mount Pleasant, SC: Hawkes Learning, 2016.

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First Fundamental Theorem of Calculus

## Cite this as:

Weisstein, Eric W. "First Fundamental Theorem of Calculus." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstFundamentalTheoremofCalculus.html