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Unformatted text preview: 07.01.1 Chapter 07.01 Primer on Integration After reading this chapter, you should be able to: 1. define an integral, 2. use Riemann’s sum to approximately calculate integrals, 3. use Riemann’s sum and its limit to find the exact expression of integrals, and 4. find exact integrals of different functions such as polynomials, trigonometric function and transcendental functions. What is integration? The dictionary definition of integration is combining parts so that they work together or form a whole. Mathematically, integration stands for finding the area under a curve from one point to another. It is represented by ∫ b a dx x f ) ( where the symbol ∫ is an integral sign, and a and b are the lower and upper limits of integration, respectively, the function f is the integrand of the integral, and x is the variable of integration. Figure 1 represents a graphical demonstration of the concept. Riemann Sum Let f be defined on the closed interval ] , [ b a , and let ∆ be an arbitrary partition of ] , [ b a such as: b x x x x x a n n = < < < < < = − 1 2 1 ..... , where i x ∆ is the length of the th i subinterval (Figure 2). If i c is any point in the th i subinterval, then the sum ∑ = − ≤ ≤ ∆ n i i i i i i x c x x c f 1 1 , ) ( 07.01.2 Chapter 07.01 is called a Riemann sum of the function f for the partition ∆ on the interval ] , [ b a . For a given partition ∆ , the length of the longest subinterval is called the norm of the partition. It is denoted by ∆ (the norm of ∆ ). The following limit is used to define the definite integral. Figure 1 The definite integral as the area of a region under the curve, ∫ = b a dx x f Area ) ( . If i c is any point in the th i subinterval, then the sum i i i i n i i x c x x c f ≤ ≤ ∆ − = ∑ 1 1 , ) ( Figure 2 Division of interval into n segments. is called a Riemann sum of the function f for the partition ∆ on the interval ] , [ b a . For a given partition ∆ , the length of the longest subinterval is called the norm of the partition. It is denoted by ∆ (the norm of ∆ ). The following limit is used to define the definite integral. ∑ = → ∆ = ∆ n i i i I x c f 1 ) ( lim x x 1 ... x i1 x i … x n1 x n ∆ x i x y y = f(x) a b Primer on Integration 07.01.3 This limit exists if and only if for any positive number ε , there exists a positive number δ such that for every partition ∆ of ] , [ b a with δ < ∆ , it follows that ε < ∆ − ∑ = n i i i x c f I 1 ) ( for any choice of i c in the th i subinterval of ∆ . If the limit of a Riemann sum of f exists, then the function f is said to be integrable over ] , [ b a and the Riemann sum of f on ] , [ b a approaches the number I ....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida  Tampa.
 Spring '08
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