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Second Fundamental Theorem of Calculus

In the most commonly used convention (e.g., Apostol 1967, pp. 205-207), the second fundamental theorem of calculus, also termed "the fundamental theorem, part II" (e.g., Sisson and Szarvas 2016, p. 456), states that if f is a real-valued continuous function on the closed interval [a,b] and F is the indefinite integral of f on [a,b], then

int_a^bf(x)dx=F(b)-F(a).

This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.

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.

See also

Derivative, First Fundamental Theorem of Calculus, Fundamental Theorems of Calculus, Integral

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References

Apostol, T. M. "Primitive Functions and the Second Fundamental Theorem of Calculus." §5.3 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 205-207, 1967.Edwards, J. "The Fundamental Proposition." §17 in A Treatise on the Integral Calculus with Applications, Examples and Problems. New York: Chelsea, pp. 12-14, 1954.Sisson, P. and Szarvas, T. Single Variable Calculus with Early Transcendentals. Mount Pleasant, SC: Hawkes Learning, 2016.

Cite this as:

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

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