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It's a lousy picture, but it is trying to show that Ln (the y value of each rectangle is the y value of the curve on the left side of the rectangle) is less than the area under the curve, because the curve is "increasing continuous function".
The area Rn (each rectangle takes the y-value of the curve from its right side) is greater than A.
so (1) Ln<A<Rn
The picture also tries to show that each small rectangle formed by the difference in the y-values of the Left limit and the right limit can be stacked up to produce a tall thin rectangle. It has width (b-a)/n and height f(b)-f(a)
so
(2) Rn-Ln = (b-a) ( f(b)- f(a))/n
from (1) we have Ln < A, or Ln-Rn<A-Rn
multiply both sides by -1 and flip the relation:
Rn-Ln> Rn-A
from (2), we can substitute for Rn-Ln to get
(3) Rn-A < (b-a) ( f(b)-f(a))/n