Please look at the proof's that are in these links.
I came across this proof for why area between a function and the x-axis is equal to the definite integral. However I do not understand it. It approximated the area by splitting the area up into rectangles of width x(i-1)- xi and height f'(c), using the mean value theorem. However, I do not understand this proof as the area is equal the definite integral however many rectangles are used. So if 2 rectangles are used the approximation of the area under the curve would still be the definite integral. This does not seem intuitive to me as the area should surely be different if the number of rectangles changes. Surely, the the area should only be equal to the definite integral when the number of rectangles tends to infinity and the width's of the rectangles tend to 0.
Could somebody please explain this to me.