I have attached a picture below, and the thought of this idea is confusing me too much.
-> In the first graph, I have taken the area of a single rectangle (say, first rectangle) as M * Δx , where M represents the arithmetic mean of f(x) and f(x) + Δy ( Δy is f(x+Δx) - f(x) ) which i thought would givea better approximation of the area as opposed to directly taking area as f(x)Δx
So the question,(please see the pic first) why dont we take the area under a curve as ∫ (f(x) + dy/2)dx? because when Δx is big, the expression of area under a curve as ∫ (f(x) + Δy/2)Δx would give a more precise result. will it give the same as Δx approaches 0?
Please tell where am i going wrong.
p.s. - I am a newbie at calculus, so bear with me if my question is stupid