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:DRf:D\rightarrow\mathbb{R} over the closed interval I=[a,b]DI=[a,b]\subset D with partition P={[x0,x1),...,[xn1,xn]}P=\{[x_0,x_1),...,[x_{n-1},x_n]\}is S=i=1nf(ξi)(xixi1),ξ[xi1,xi].S=\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1}),\, \xi\in[x_{i-1},x_i]. Where a=x0<x1<x2...<xn=ba=x_0<x_1<x_2...<x_n=b and ξi\xi_i is a random number from the interval [xi1,xi][x_{i-1},x_i].

Some Riemann sums have special names, e.g. the left Riemann sum is a sum where ξi=xi1\xi_i=x_{i-1} and the right Riemann sum is a sum where ξi=xi\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(ξi)f(\xi_i) in our sum with sup/inf{f(x),x[xi1,xi]}\sup/\inf \{f(x), x\in[x_{i-1},x_i]\}. All this definitons help us finaly derive the notion of Riemann integral abf(x)dx=limn(i=1nf(ξi)(xixi1)).\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
7 years, 2 months ago

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