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n f 1 f n f 1 f n x 1 X f   n b a b a k 1 
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This sum speci es the average value of the n functions at the midpoints of the intervals. However, we may abstract the last member of the string of equalities (dropping the special conditions) and de ne b n 1 X 1 lim f   f x dx n!1 b a b a a k 1 as the average value of f on a; b .
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Of course, the question of for what value x  the average is attained is not answered; and, in fact, in general, only existence not the value can be demonstrated. To see that there is a point x  such that f  represents the average value of f on a; b , recall that a continuous function on a closed interval has maximum and minimum values, M and m, respectively. Thus (think of the integral as representing the area under the curve). (See Fig. 5-2.)
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y = f (x) M D m A a b_a B b x C
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or 1 m@ b a b
f x dx @ M
Since f is a continuous function on a closed interval, there exists a point x  in a; b intermediate to m and M such that f  1 b a b f x dx
While this example is not a rigorous proof of the rst mean value theorem, it motivates it and provides an interpretation. (See 3, Theorem 10.) 1. First mean value theorem. If f x is continuous in a; b , there is a point  in a; b such that b
f x dx b a f 
2. Generalized rst mean value theorem. If f x and g x are continuous in a; b , and g x does not change sign in the interval, then there is a point  in a; b such that b f x g x dx f 
b g x dx 5
This reduces to (4) if g x 1.
INTEGRALS
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CONNECTING INTEGRAL AND DIFFERENTIAL CALCULUS In the late seventeenth century the key relationship between the derivative and the integral was established. The connection which is embodied in the fundamental theorem of calculus was responsible for the creation of a whole new branch of mathematics called analysis. De nition: Any function F such that F 0 x f x is called an antiderivative, primitive, or inde nite integral of f . The antiderivative of a function is not unique. constant c This is clear from the observation that for any
F x c 0 F 0 x f x The following theorem is an even stronger statement. Theorem. Any two primitives (i.e., antiderivatives), F and G of f di er at most by a constant, i.e., F x G x C. (See the problem set for the proof of this theorem.)
EXAMPLE. x3 If F 0 x x2 , then F x x2 dx c is an inde nite integral (antiderivative or primitive) of x2 . 3
The inde nite integral (which is a function) may be expressed as a de nite integral by writing x f x dx f t dt
The functional character is expressed through the upper limit of the de nite integral which appears on the right-hand side of the equation. This notation also emphasizes that the de nite integral of a given function only depends on the limits of integration, and thus any symbol may be used as the variable of integration. For this reason, that variable is often called a dummy variable. The inde nite integral notation on the left depends on continuity of f on a domain that is not described. One can visualize the de nite integral on the right by thinking of the dummy variable t as ranging over a subinterval c; x . (There is nothing unique about the letter t; any other convenient letter may represent the dummy variable.) The previous terminology and explanation set the stage for the fundamental theorem. It is stated in two parts. The rst states that the antiderivative of f is a new function, the integrand of which is the derivative of that function. Part two demonstrates how that primitive function (antiderivative) enables us to evaluate de nite integrals.
THE FUNDAMENTAL THEOREM OF THE CALCULUS Part 1 Let f be integrable on a closed interval a; b . Let c satisfy the condition a @ c @ b, and de ne a new function x F x f t dt if a @ x @ b
Then the derivative F x exists at each point x in the open interval a; b , where f is continuous and F 0 x f x . (See Problem 5.10 for proof of this theorem.) Part 2 As in Part 1, assume that f is integrable on the closed interval a; b and continuous in the open interval a; b . Let F be any antiderivative so that F 0 x f x for each x in a; b . If a < c < b, then for any x in a; b x f t dt F x F c
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