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Chapter 5 Integrals 5.2 The Definite Integral In this handout: Riemann sum Definition of a definite integral Properties of the definite integral

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Riemann Sum An ordered collection P=(x 0,x 1,…,x n ) of points of a closed interval I = [a,b] satisfying a = x 0 < x 1 < …< x n-1 < x n = b is a partition of the interval [a,b] into subintervals I k =[x k-1,x k ]. Let Δx k = x k -x k-1 For a partition P=(x 0,x 1,…,x n ), let |P| = max{ Δ x k, k=1,…,n}. The quantity |P| is the length of the longest subinterval I k of the partition P. a = x 0 x1x1 x2x2 x n-1 x n =b |P||P| Choose a sample point x i * in the subinterval [x k-1,x k ]. A Riemann sum associated with a partition P and a function f is defined as:

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Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) Note that the integral does not depend on the choice of variable. If f is a function defined on [a, b], the definite integral of f from a to b is the number provided that this limit exists. If it does exist, we say that f is integrable on [a, b]. Definition of a Definite Integral

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Theorem: If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is integrable on [a, b]. If f is integrable on [a, b], then in calculating the value of an integral we are free to choose the partitions and sample points to simplify the calculations. It is often convenient to take a regular partition; that is, all the subintervals have the same length Δx. Existence of a Definite Integral

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If the upper and lower limits are equal, then the integral is zero. Reversing the limits changes the sign. Constant multiples can be moved outside. Properties of the Integral where c is any constant Integrals can be added (or subtracted). Intervals can be added (or subtracted.)

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If f(x) ≥ g(x) for a ≤ x ≤ b, then If f(x) ≥ 0 for a ≤ x ≤ b, then If m ≤ f(x) ≤ M for a ≤ x ≤ b, then Comparison Properties of the Integral

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