I would like to rephrase. By saying that i took the area of the rectangle as (f(x)+Δy/2)Δx in the first graph , i meant to say that i took the height of an individual rectangle as [f(x)+f(x+Δx)]/2 ( not as f(x), neither as f(x+Δx) )to get the mean height, or when Δx is big, a better approximation of the area as Σ [{f(x)+f(x+Δx)}/2]*Δx
Again rephrasing, why dont we take the area under a curve as ∫ {f(x)+f(x+dx)}/2*dx? because when Δx is big, the expression for area under a curve as Σ [{f(x)+f(x+Δx)}/2]*Δx would give a more precise result. Will it give the same as Δx approaches zero?