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is called a Riemann sum for f on [a� b] and is denoted by n � f (tk )Δxk . k=1 Remark 5.1.1. Note that if f is a continuous function on [a� b] and P = �x0 � x1 � x2 � . . . � xn } is a partition of [a� b], then Lf (P ) ≤ n � f (tk )Δxk ≤ Uf (P ) k=1 for every number tk in [xk−1 � xk ], k = 1� 2� . . . � n. In particular, if f (x) ≥ 0 on [a� b], then the area A of the graph of f on [a� b] satisfies Lf (P ) ≤ A ≤ Uf (P ) for every partition P = �x0 � x1 � x2 � . . . � xn } of [a� b]. Example 5.1.1. Let f be a continuous function on [a� b] and let P = �x0 � x1 � x2 � . . . � xn } be a partition of [a� b]. If for each integer k between 1 and n, we choose tk to be the left endpoint of [xk−1 � xk ], the corresponding Riemann sum is called the left sum. If for each integer k between 1 and n, we choose tk to be the right endpoint of [xk−1 � xk ], the corresponding Riemann sum is called the right sum. If for each integer k between 1 and n, we choose tk to be the midpoint of [xk−1 � xk ], the corresponding Riemann sum is called the midpoint sum. 5.2 The definite integral Theorem 5.2.1. For any function f that is continuous on [a� b] there exists a unique number I with the following property: For any � > 0 there exists δ > 0 such that if �x0 � x1 � x2 � . . . � xn } is any partition of [a� b] each of whose subintervals has length less than � δ, and if xk−1 ≤ tk ≤ xk for each k between 1 and n, the associated Riemann sum nk=1 f (tk )Δxk satisfies |I − n � f (tk )Δxk | < �. k=1 Definition 5.2.1. Let f be a continuous function on [a� b]. The definite integral of f from a to b is the unique number I which the Riemann sums approach as �P � → 0. This number is denoted by � b f (x)dx. a 25 � is the integral sign, f is the integrand, and � b n � f (x)dx = lim f (tk )Δxk . �P �→0 a k=1 Definition 5.2.2. Let f be continuous and nonnegative on [a� b], and R the region between the graph of f and the x−axis on [a� b]. The area of R is defined to be � b f (x)dx. a Example 5.2.1. Let c be a constant. Then �b 1. a cdx = c(b − a). �b 2 2 . 2. a xdx = b −a 2 �b 3 3 . 3. a x2 dx = b −a 3 Remark 5.2.1. Let f be a continuous function on [a� b]. The error in approximating �b f (x)dx by a Riemann sum is a | 5.3 � b f (x)dx − a n � f (tk )Δxk | k=1 Special properties of the definite integral Definition 5.3.1. Let f be a continuous function on [a� b]. Then � a � a � b f (x)dx = 0� f (x)dx = − f (x)dx. a b Theorem 5.3.1. For all a� b� c, �b a a cdx = c(b − a). Theorem 5.3.2. Let a� b� c be numbers and let f be a continuous function. Then � c � b � b f (x)dx = f (x)dx + f (x)dx. a a c Definition 5.3.2. Let f be a function that is continuous on [a� b] except at finitely many numbers of the interval. Then f is called a piecewise continuous function. Theorem 5.3.3. Let f be a continuous on [a� b] and suppose that m ≤ f (x) ≤ M for all x in [a� b]. Then � b f (x)dx ≤ M (b − a). m(b − a) ≤ a 26