lect116_22rev_f11

lect116_22rev_f11 - Thursday, November 3 Lecture 22:...

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Thursday, November 3 Lecture 22: Riemann sums (Refers to Section 5.1 in your text) Students who have mastered the content of this lecture know about : Sigma notation , Riemann sums Students who have practiced the techniques presented in this lecture will be able to : Express finite sums using sigma notation , interpret correctly finite sums expressed using sigma notation , recognize Riemann sums, evaluate finite Riemann sums . 22.1 Notation Sigma notation. The sigma notation is a method of writing a sum of many terms in a more compact way. We will explain clearly how to interpret this symbol and how to use it correctly. There should be no ambiguities. The following examples provide an understanding on how we should read this symbol: 22.1.1 Closed form for the terms Normally a sum expressed in sigma notation expresses its terms in a closed form. For example, the sum below has its most succinct form on the right: Thus sum below is interpreted on the right: The following sum is defined as having this particular meaning:
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22.1.2 Three important sum formulas There are few formulas for finite sums that we may need in this section. They are: 22.1.3 Examples Compute the following finite sums: Solution : a) 4 × [ 9 × 10/2] = 45 × 4 = 180. b) 2 × [4 × 5 × (2 × 4 + 1)/6] (4 × 5/2) + 4 × 2 c) 2 × 90 = 180 d) (1/5)[3 × (4 × 5/2) (4 × 4)] = (1/5)[30 16] = 14/5 e) 2 × (5 × 6/2) = 30
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22.2 Riemann sums generated by a function f(x). Suppose we wish to approximate the area of the region bounded by the function f ( x ) = x 2 over the interval [0, 4]. We could proceed by approximating the area of the region by constructing a set of rectangles whose height is determined by the function f ( x ) as illustrated below.
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This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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lect116_22rev_f11 - Thursday, November 3 Lecture 22:...

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