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set11 - Lecture 30 The denite integral(Relevant section...

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Lecture 30 The definite integral (Relevant section from Stewart, Sixth Edition: Section 5.2) The material presented in this lecture closely followed the presentation of Section 5.2 of the text- book, pp. 366-376. Lecture 31 The Fundamental Theorem of Calculus The material presented in this lecture, including the proofs of FTC I and II, closely followed the presentation of Section 5.3 of the textbook, pp. 379-387. This is perhaps one of the most important results of this course. In both of the results stated below (and proved in class), we assume that f ( x ) is a continuous function on R . 1. First define the function g ( x ) = integraldisplay x a f ( t ) dt. (1) Then g ( x ) = f ( x ) . (2) This result is often known as the First Fundamental Theorem of Calculus , or simply “ FTC I .” 2. The above result establishes that the function g ( x ) is an antiderivative of f ( x ). The question is, “Which one?” The answer is that g ( x ) is the antiderivative of f ( x ) for which g (0) = 0. From the fact that all antiderivatives F ( x ), f ( x ) are related to g ( x ) as follows, F ( x ) = g (

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