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6.4 - The Definite Integral I. The Definite Integral If we let the number of rectangles on [a, b] increase without bound and if f is continuous on [a, b], then n lim f (ck ) x n k 1 is the definite integral of f (x) on [a, b] and we write n f (ck ) x lim n k 1 b a f (x) dx. Geometric interpretation: II. Properties: (in addition to all the previous properties of integrals) a 1. f (x) dx 0 2. a 3. If f ( x) 0, b a a b f (x) dx f (x) dx b f (x) dx 0 a 4. If a c b , then b c a a f (x) dx f (x) dx b f (x) dx . c Examples: 4 1. 2 dx 1 3 2. ( 2x 1) dx 2 More generally, (1) f can take on negative values as well as positive (2) rectangles can vary in width (3) points to compute height can be anywhere in the subintervals Riemann Sums: Sn n f (ck ) x k 1 where ck [ xk 1, xk ] .