Question

Riemann theorem:
Let f(x) =x^2, and let P denote a partition of an interval [a,b]
i) Let L(f,P), M(f,P)and R(f,P) denote respectively the left end point, the midpoint and the right endpoint sums of f(x) on P. Show that L+4M+R = 2(b^3-a^3). I got this part
ii) Show that for any partition P with \(||P||<\dfrac{\varepsilon}{b^2-a^2}, R-L<\varepsilon\).
Please, help

Answer 1

@freckles

Answer 2

My attempt:
\(R(f,P, x_i) = \sum_{i=1}^n f(x_i)||P||\)
\(L(f,P, x_{i-1})= \sum_{i=1}^n f(x_{i-1})||P||\)
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\(R-L = \sum_{i=1}^n(f(x_i)-f(x_{i-1})||P||\)
Let \(\delta =\dfrac{\varepsilon}{b^2-a^2}\), then \(||P||<\delta\)

Answer 3

and \((R-L) <\delta \sum_{i=1}^n (x_i^2-x_{i-1}^2)=\varepsilon\) on [a,b]

Answer 4

iii) from i) and ii) conclude that for any P with \(||P||< \dfrac{\varepsilon}{b^2-a^2}\) and for any choice of test points \(x_i^*, x_i^#\in [x_{i-1}, x_i] |R(f,P,x_i^*)-R(f,P,x_i^#)|<\varepsilon\)

Answer 5

I doubt myself for this part, since it is trivial. My argument: if we pick any point in \([x_{i-1},x_i]\), we make a "finerment" partition. Hence for the 2 new points , we have 2 news intervals. The ||P|| works well for the new one. What are we supposed to do then?