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.
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 $\displaystyle R_n - L_n = \Delta x (f(b)-f(a))$.