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Notes 7FT (Second Theorem of Calculus)

Notes 7FT (Second Theorem of Calculus) - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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FT. SECOND FUNDAMENTAL THEOREM 1. The Two Fundamental Theorems of Calculus The Fundamental Theorem of Calculus really consists of two closely related theorems, usually called nowadays (not very imaginatively) the First and Second Fundamental Theo - rems. Of the two, it is the First Fundamental Theorem that is the familiar one used all the time. It is the theorem that tells you how to evaluate a definite integral without having to go back to its definition as the limit of a sum of rectangles. First Fundamental Theorem Let f(x) be continuous on [a, b]. Suppose there is a function F(x) such that f(x) = F'(x) . Then b b (1) f f()dx F(b) F(a). = ) = - F(x) (The last equality just gives another way of writing F(b) - F(a) that is in widespread use.) Still another way of writing the theorem is to observe that F(x) is an antiderivative for f(x), or as it- is sometimes called, an indefinite integral for f(x); using the standard notation for indefinite integral and the bracket notation given above, the theorem would be written (1') f (x) dx = f (x)d . Writing the theorem this way makes it look sort of catchy, and more importantly, it avoids having to introduce the new symbol F(x) for the antiderivative. In contrast with the above theorem, which every calculus student knows, the Second Fundamental Theorem is more obscure and seems less useful. The purpose of this chapter is to explain it, show its use and importance, and to show how the two theorems are related. To start things off, here it is. Second Fundamental Theorem. Let f(x) be continuous, and fix a. (2) Set F() = f (t) dt; then F'(x) = f(x). We begin by interpreting (2) geometrically. Start with the graph of f(t) in the ty-plane. Then F(x) represents the area under f(t) between a and x; it is a function' of x. Its derivative - the rate of change of area with respect to x - is the length of the dark vertical line. This is what (2) says e met.ricrnll ISimmons calls this function A(x) on p. 207 (2nd edition); this section of Simmons is another presentation of much of the material given here.
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Example 1. Verify (2) if f(x) = 2xsin 2 and a= 0. Solution. Here we can integrate explicitly by finding an antiderivative (using the first fundamental theorem): F(W) = 2t sintdt - cos t2 -cosx + 1; differentiating by the chain rule, we verify that indeed F'(x) = 2zsinx 2 , as predicted by (2). O *Example 2. Let F(z) = Ssint dt. Find F'(7r/2). Solution. Neither integration techniques nor integral tables will produce an explicit antiderivative for the function in the integrand. So we cannot use (1). But we can use (2), which says that sin 7r/2 1 F'(R/2) = i/2 /2 Many students feel the Second Fundamental Theorem is "obvious"; these students are confusing it with the similar-looking (2') Let F(x) = f f(x) dx; then F'(x) = f(x).
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