RIEMANN INTEGRAL

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence

 a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!

Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left.

This partitions the interval [a,b] into n sub-intervals [xi−1, xi] indexed by i, each of which is “tagged” with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as

\sum_{i=1}^{n} f(t_i) \Delta_i ;

thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xixi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1…n Δi. The Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have
<img src="http://upload.wikimedia.org/math/5/b/c/5bcae45e90b157b48189d638bdb0d244.png&quot; alt="\left| S – \sum_{i=1}^{n} f(t_i)\Delta_i \right|

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.

About NICO MATEMATIKA

Welcome to my blog. My name is Nico. Admin of this blog. I am a student majoring in mathematics who dreams of becoming a professor of mathematics. I live in Kwadungan, Ngawi, East Java. Hopefully in all the posts I can make a good learning material to the intellectual life of the nation. After the read, leave a comment. I always accept criticism suggestion to build a better me again .. Thanks for visiting .. : mrgreen:

Posted on August 29, 2011, in education and tagged , , , , , , , . Bookmark the permalink. 1 Comment.

  1. nice blog, nico.. ^^

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