f is contin and increasing on[a,b] show that Rieman sum Rn decrease with n and Ln increase with n

Question

Rn=right riemann sum

Ln=left riemann sum.

thanks

Answer 1

As f is increasing over the interval [a,b] we can state that f(x+δx)>f(x), where δx=(b-a)/n.

L_n takes two consecutive values f(x+δx) and f(x) and uses the rectangular area f(x)δx, while R_n uses the rectangular area f(x+δx)δx, therefore f(x+δx)δx>f(x)δx. It follows then that R_n>L_n when the areas are summed over the interval. As n increases δx→0, therefore f(x+δx)δx→f(x)δx, implying that R_n decreases as L_n increases.