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Stewart6e5_2

# Stewart6e5_2 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 5.2, part I: The Definite Integral, a Definition Let’s make precise all of the whatnot we’ve been doing with areas, with a Definition. Let f be a function defined on the interval [ a, b ] and divide the interval into n equal-length subintervals, each Δ x = b - a n long, with endpoints x 0 , x 1 , ..., x n , numbered from left to right. (This means that x i = a + i Δ x .) From the i th subinterval [ x i - 1 , x i ] select a point , x * i . We then define the of f from a to b , by b a f ( x ) dx = lim n →∞ n i =1 f ( x * i x if this limit exists as a real number, in which case f is said to be . The notation here is due to our old friend , and represents an elongated sum. The numbers a and b are respectively called the and limits of integration for the integral, and the function f is called the integral’s . Even though we’ve used the variable x here, that variable effectively “disap- pears” when the integral is evaluated, since the result is simply a number. For this reason, x is often called a

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Stewart6e5_2 - MATH 191 Sections 1 and 3 Calculus I Fall...

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