Stewart6e5_2part2

Stewart6e5_2part2 - MATH 191, Sections 1 and 3 Calculus I...

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Calculus I Fall 2007 Section 5.2, part II: More on Definite Integrals Recall that R b a f ( x ) dx = lim n →∞ n X i =1 f ( x * i x , where x * i is a sample point chosen from the subinterval [ x i - 1 , x i ]. Our computations using Mathematica have shown us that in selecting x * i to be the midpoint x i = x i - 1 + x i 2 of the interval [ x i - 1 , x i ], we obtain a better estimate than if we use either endpoint. This is the basis of the so-called Midpoint Rule. R b a f ( x ) dx n X i =1 f ( x i x . Very often this estimate is good enough for whatever purpose it’s needed. Example. Use the Midpoint Rule with n = 4 to estimate the integral R 3 1 1 x dx . How far off are we, according to Mathematica ? Before we move on, we should point out a number of properties of the definite integral, most of which follow from basic computations and the definition of the integral. In the properties below, let c be any constant, and let f and g both be integrable on the interval [
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Stewart6e5_2part2 - MATH 191, Sections 1 and 3 Calculus I...

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