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Unformatted text preview: THE DEFINITE INTEGRAL We saw a limit of the form lim n [ f ( x * 1 ) x + f ( x * 2 ) x + + f ( x * n ) x ] = lim n n X i =1 f ( x * i ) x Because this form arises frequently in a wide variety of situations, we give this type of limit a special name and notation. Definition 1. If f is a continuous function defined for a x b , we divide the interval [ a,b ] into n subintervals of equal width x = ( b- a ) /n . We let x = a,x 1 ,x 2 , ,x n = b be the endpoints of these subintervals and we 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 X i =1 f ( x * i ) x The symbol R is called an integral sign . In the notation Z b a f ( x ) dx , f ( x ) is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit . The procedure of calculating an integral is called....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
- Fall '09