Why is it that R_{n}-L_{n}= $\displaystyle \Delta x$(f(b)-f(a)) on [a,b]? I've tried manipulating summations but I can't seem to figure it out.

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- Aug 25th 2012, 06:37 PMfeifanDifference of right and left Riemann sum
Why is it that R

_{n}-L_{n}= $\displaystyle \Delta x$(f(b)-f(a)) on [a,b]? I've tried manipulating summations but I can't seem to figure it out. - Aug 25th 2012, 11:36 PMVlasevRe: Difference of right and left Riemann sum
The simplest way is to say that except for the first summand of $\displaystyle L_n$ (it is $\displaystyle f(a)\Delta x$) and the last summand of $\displaystyle R_n$ (it is $\displaystyle f(b)\Delta x$), all the other terms appear in both sums. When you subtract, you are left with $\displaystyle R_n - L_n = \Delta x (f(b)-f(a))$.