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Playing with Integrals: Riemann sums

When we introduce definite integral in Mathematics we usually start with very formal definition -- the Riemann Sum. Many of us consider it quite dull and boring, because it doesn't directly help in the pen-and-paper calculation (but it's basis for computer calculation) of integrals. However, understanding this definition and taking advantage of it is extremelly useful when tackling an Olympiad problem.

The Riemann sum of a function \(f:D\rightarrow\mathbb{R}\) over the closed interval \(I=[a,b]\subset D\) with partition \(P=\{[x_0,x_1),...,[x_{n-1},x_n]\}\)is \[S=\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1}),\, \xi\in[x_{i-1},x_i].\] Where \(a=x_0<x_1<x_2...<x_n=b\) and \(\xi_i\) is a random number from the interval \([x_{i-1},x_i]\).

Some Riemann sums have special names, e.g. the left Riemann sum is a sum where \(\xi_i=x_{i-1}\) and the right Riemann sum is a sum where \(\xi_i=x_i\). The last two types of sums I want to discuss are upper and lower Riemann sums. To get them we subtitute \(f(\xi_i)\) in our sum with \(\sup/\inf \{f(x), x\in[x_{i-1},x_i]\}\). All this definitons help us finaly derive the notion of Riemann integral \[\int^b_a f(x)\,dx=\lim_{n\to\infty}\left(\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\right).\]

Note by Nicolae Sapoval
3 years ago

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