# fundamental theorem of calculus

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## fundamental theorem of calculus

[¦fən·də¦ment·əl ¦thir·əm əv ′kal·kyə·ləs]
(mathematics)
Given a continuous function ƒ(x) on the closed interval [a,b ] the functional is differentiable on [a,b ] and F(x) = ƒ(x) for every x in [a,b ], and if G is any function on [a,b ] such that G ′(x) = ƒ(x) for all x in [a,b ], then
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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So we can see that the Fundamental Theorem of Calculus applies and we can restate the above integration by parts formula as follows.
Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.
 Bayoumi A., 1999, Fundamental theorem of calculus for locally bounded spaces, J.
While they don't offer a proof of the Fundamental Theorem of Calculus, they do a decent job of making clear, in English, what it means.
However, since g is differentiable over [0,1], it is reasonable to expect from the Fundamental Theorem of Calculus that [[integral].sup.1.sub.0] g' = g(1) - g(0), the integral [[integral].sup.1.sub.0] g' should depend on only the value of g(1) - g(0) regardless if g' is integrable.
By the fundamental theorem of calculus, for all y [member of] [0, T] we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore
The First Fundamental Theorem of Calculus is not easily grasped by students: From experience, it is very hard for students taking up calculus for the first time to understand the equation

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