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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 5.2, part I: The Definite Integral, a Definition Lets make precise all of the whatnot weve 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 x = b- a n long, with endpoints x , x 1 , ..., x n , numbered from left to right. (This means that x i = a + i 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 Z b a f ( x ) dx = lim n n X 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 integrals . Even though weve used the variable x here, that variable effectively disap- pears when the integral is evaluated, since the result is simply a number. Forpears when the integral is evaluated, since the result is simply a number....
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This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.
- Fall '07