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Unformatted text preview: Section 5.2: The Definite Integral Instructor: Ms. Hoa Nguyen (email@example.com) Definition of a Definite Integral f is a continuous function defined for a x b . Divide [ a, b ] into n subintervals whose width is: 4 x = b- a n . n subintervals are: [ x , x 1 ] , [ x 1 , x 2 ] , [ x 2 , x 3 ] , , [ x n- 1 , x n ] where x = a and x n = b . Let x * 1 , x * 2 , , x * n be any sample points in these subintervals, so x * i lies in the i th subinterval [ x i- 1 , x i ]. Then the definite integral of f from a to b is Z b a f ( x ) dx = lim n n i =1 f ( x * i ) 4 x Remarks : Because we have assumed that f is continuous, it can be proved that the limit in the above definition always exists and gives the same value no matter how we choose the sample points x * i . The above limit also exists if f has a finite number of removable or jump discontinuities (but not infinite discontinuities)....
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.
- Fall '08