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MTH 252 Integral Calculus Chapter 6 – Integration Section 6.5 – The Definite Integral Copyright © 2005 by Ron Wallace, all rights reserved.

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Partitions ax0ax0 bxnbxn x1x1 x3x3 x2x2 x n-1 … A partition of [a, b] is a collection of n+1 points P n = {x 0, x 1, x 2, x 3, …, x n-1, x n } such that a = x 0 < x 1 < x 2 < x 3 < … < x n-1 < x n = b. Subinterval Length (k th interval): x k = x k - x k-1 Mesh Size of a Partition: max x k = largest subinterval length Regular Partition: x k = x = (b-a)/n, for all k [i.e. all of the subintervals have the same length]

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Riemann Sum If f(x) is continuous over [a,b], P n is a partition of [a,b], and x k * is any point in the k th subinterval, then is called a Riemann Sum.

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Definite Integral If exists, then f(x) is integrable and the limit is the “definite integral of f(x) from a to b.”

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Definite Integral Equivalent, simplified, and more practical form. Since any partition can be used, use a regular partition. Since x k * is any point in the k th interval, use x k (the right endpoint). Therefore … where …

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Definite Integrals & Area Problems If f(x) ≥ 0 over [a,b], then is the area under the curve. If f(x) ≤ 0 over [a,b], then is the area above the curve. Therefore, geometry can be used to determine some integrals. Example:

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Two extensions to the definition.

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Properties of Definite Integrals This is true if a < c < b, c < a < b, or a < b < c.

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