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# 5_2 - Section 5.2 The Denite Integral Instructor Ms Hoa...

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Section 5.2: The Definite Integral Instructor: Ms. Hoa Nguyen Definition of a Definite Integral f is a continuous function defined for a x b . Divide [ a, b ] into n subintervals whose width is: x = b - a n . n subintervals are: [ x 0 , x 1 ] , [ x 1 , x 2 ] , [ x 2 , x 3 ] , ·· · , [ x n - 1 , x n ] where x 0 = 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 b a f ( x ) dx = lim n →∞ Σ n i =1 f ( x * i ) 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). The symbol is called an integral sign (introduced by Leibniz). f ( x ) in b a f ( x ) is called the integrand . a and b are called the limits of integration . a is the lower limit . b

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5_2 - Section 5.2 The Denite Integral Instructor Ms Hoa...

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